#72341098

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
prime radical	#251911	C₂
LCS	G ▷ C₂⁵ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂² ▷ C₁

2247 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)(3,11)(4,15,17,8)(5,9)(6,10)(7,14)(12,21,29,23)(13,22)(16,24,19,25)(18,27)(20,26)(28,31,30,32)(33,34)(35,39)(36,40,48,56)(37,49,50,41)(38,42)(43,54,46,51)(44,57,59,52)(45,53)(47,55)(58,64)(60,63)(61,62);(1,3)(2,7)(4,12,17,29)(5,18)(6,14)(8,23,15,21)(9,22)(10,11)(13,20)(16,30,19,28)(24,32,25,31)(26,27)(33,35)(34,39)(36,46,48,43)(37,44,50,59)(38,45)(40,54,56,51)(41,57,49,52)(42,53)(47,58)(55,64)(60,61)(62,63);(1,4)(2,8)(3,12,14,29)(5,19,20,16)(6,17)(7,21,11,23)(9,25,26,24)(10,15)(13,30)(18,28)(22,32)(27,31)(33,36,38,48)(34,40)(35,43,45,46)(37,47)(39,51)(41,55,49,62)(42,56)(44,58)(50,61)(52,63,57,64)(53,54)(59,60);(1,5)(2,9,10,26)(3,13,14,18)(4,16)(6,20)(7,22)(8,24,15,25)(11,27)(12,28,29,30)(17,19)(21,31)(23,32)(33,37)(34,41)(35,44,45,59)(36,47,48,61)(38,50)(39,52,53,57)(40,55,56,62)(42,49)(43,58)(46,60)(51,63)(54,64);(1,3)(2,7)(4,12)(5,18)(6,14)(8,23)(9,22)(10,11)(13,20)(15,21)(16,28)(17,29)(19,30)(24,31)(25,32)(26,27)(34,42)(36,48)(39,53)(40,56)(44,59)(47,61)(52,57)(58,60)

Polycyclic presentation:

2^10|a^2=g,b^2=g,c^2=fghj,d^2=fg,b^a=bgi,c^a=ch,c^b=cf,d^a=dfhi,d^b=dh,d^c=dg,e^a=egij,e^b=eg,e^c=eghij,e^d=egh,f^a=fi,f^b=fij,f^c=fi,f^d=fj,f^e=fi,g^c=gij,g^d=gj,h^a=hij,h^b=hj,h^c=hj,h^d=hij

Conjugacy classes & character table

61 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72341103, #72341110, #72341113
maximal normal subgroup of	#72341103, #72341110, #72341113

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.7);;
gap> SetPower(coll, 2, F.7);;
gap> SetPower(coll, 3, F.6*F.7*F.8*F.10);;
gap> SetPower(coll, 4, F.6*F.7);;
gap> SetConjugate(coll, 2, 1, F.2*F.7*F.9);;
gap> SetConjugate(coll, 3, 1, F.3*F.8);;
gap> SetConjugate(coll, 3, 2, F.3*F.6);;
gap> SetConjugate(coll, 4, 1, F.4*F.6*F.8*F.9);;
gap> SetConjugate(coll, 4, 2, F.4*F.8);;
gap> SetConjugate(coll, 4, 3, F.4*F.7);;
gap> SetConjugate(coll, 5, 1, F.5*F.7*F.9*F.10);;
gap> SetConjugate(coll, 5, 2, F.5*F.7);;
gap> SetConjugate(coll, 5, 3, F.5*F.7*F.8*F.9*F.10);;
gap> SetConjugate(coll, 5, 4, F.5*F.7*F.8);;
gap> SetConjugate(coll, 6, 1, F.6*F.9);;
gap> SetConjugate(coll, 6, 2, F.6*F.9*F.10);;
gap> SetConjugate(coll, 6, 3, F.6*F.9);;
gap> SetConjugate(coll, 6, 4, F.6*F.10);;
gap> SetConjugate(coll, 6, 5, F.6*F.9);;
gap> SetConjugate(coll, 7, 3, F.7*F.9*F.10);;
gap> SetConjugate(coll, 7, 4, F.7*F.10);;
gap> SetConjugate(coll, 8, 1, F.8*F.9*F.10);;
gap> SetConjugate(coll, 8, 2, F.8*F.10);;
gap> SetConjugate(coll, 8, 3, F.8*F.10);;
gap> SetConjugate(coll, 8, 4, F.8*F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);