#72341012

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#46996
commutator, Frattini, cosocle	C₂⁵	C₂⁵
LCS	G ▷ C₂⁵ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂² ▷ C₁

2362 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 64 points:

(1,2)(3,13,15,8)(4,9,17,23)(5,10)(6,20)(7,12)(11,22)(14,28,21,24)(16,25,31,29)(18,30)(19,27)(26,32)(33,34)(35,44)(36,41,49,56)(37,42,50,57)(38,40)(39,43)(45,52,47,55)(46,58,51,53)(48,54)(59,62)(60,63)(61,64);(1,3)(2,8)(4,5)(6,21,22,14)(7,15)(9,10)(11,28,20,24)(12,13)(16,18,31,32)(17,19)(23,27)(25,26,29,30)(33,35)(34,40,43,44)(36,47)(37,46,50,51)(38,39)(41,52,56,55)(42,53)(45,49)(48,60,61,59)(54,62)(57,58)(63,64);(1,4,7,17)(2,9,12,23)(3,5,15,19)(6,16)(8,10,13,27)(11,29)(14,18)(20,25)(21,32)(22,31)(24,30)(26,28)(33,36,39,49)(34,41)(35,45,38,47)(37,48)(40,52)(42,54,57,64)(43,56)(44,55)(46,60)(50,61)(51,59)(53,63,58,62);(1,5)(2,10)(3,4,15,17)(6,18,22,32)(7,19)(8,23,13,9)(11,30,20,26)(12,27)(14,31)(16,21)(24,29)(25,28)(33,37)(34,42,43,57)(35,46)(36,48,49,61)(38,51)(39,50)(40,53,44,58)(41,54)(45,59,47,60)(52,62)(55,63)(56,64);(1,6,7,22)(2,11,12,20)(3,14)(4,16)(5,18,19,32)(8,24)(9,25)(10,26,27,30)(13,28)(15,21)(17,31)(23,29)(33,38,39,35)(34,40,43,44)(36,45)(37,46)(41,55)(42,53)(47,49)(48,59,61,60)(50,51)(52,56)(54,62,64,63)(57,58)

Polycyclic presentation:

2^10|a^2=f,b^2=g,c^2=gij,d^2=fg,e^2=fij,b^a=bh,c^a=cfghi,c^b=cgj,d^a=dg,d^b=dhij,d^c=dfhi,e^a=efi,e^b=efgij,e^c=eh,e^d=efghi,f^b=fi,f^c=fij,f^d=fj,g^a=gj,g^d=gj,g^e=gi,h^a=hi,h^b=hj,h^c=hi,h^d=hi,h^e=hij

Conjugacy classes & character table

55 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72341133, #72341188
maximal normal subgroup of	#72341133, #72341188

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:

gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e", "f", "g", "h", "i", "j");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 1, F.6);;
gap> SetPower(coll, 2, F.7);;
gap> SetPower(coll, 3, F.7*F.9*F.10);;
gap> SetPower(coll, 4, F.6*F.7);;
gap> SetPower(coll, 5, F.6*F.9*F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.8);;
gap> SetConjugate(coll, 3, 1, F.3*F.6*F.7*F.8*F.9);;
gap> SetConjugate(coll, 3, 2, F.3*F.7*F.10);;
gap> SetConjugate(coll, 4, 1, F.4*F.7);;
gap> SetConjugate(coll, 4, 2, F.4*F.8*F.9*F.10);;
gap> SetConjugate(coll, 4, 3, F.4*F.6*F.8*F.9);;
gap> SetConjugate(coll, 5, 1, F.5*F.6*F.9);;
gap> SetConjugate(coll, 5, 2, F.5*F.6*F.7*F.9*F.10);;
gap> SetConjugate(coll, 5, 3, F.5*F.8);;
gap> SetConjugate(coll, 5, 4, F.5*F.6*F.7*F.8*F.9);;
gap> SetConjugate(coll, 6, 2, F.6*F.9);;
gap> SetConjugate(coll, 6, 3, F.6*F.9*F.10);;
gap> SetConjugate(coll, 6, 4, F.6*F.10);;
gap> SetConjugate(coll, 7, 1, F.7*F.10);;
gap> SetConjugate(coll, 7, 4, F.7*F.10);;
gap> SetConjugate(coll, 7, 5, F.7*F.9);;
gap> SetConjugate(coll, 8, 1, F.8*F.9);;
gap> SetConjugate(coll, 8, 2, F.8*F.10);;
gap> SetConjugate(coll, 8, 3, F.8*F.9);;
gap> SetConjugate(coll, 8, 4, F.8*F.9);;
gap> SetConjugate(coll, 8, 5, F.8*F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);