#72341010

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

2310 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)(4,8)(5,18,21,9)(6,10)(7,14)(12,24)(13,23,19,28)(15,22)(16,25,20,27)(17,26)(29,32,30,31)(33,34)(35,39)(36,40,47,55)(37,48)(38,42)(41,51)(43,58,60,52)(44,53)(45,54)(46,56,50,57)(49,61)(59,63,62,64);(1,3)(2,7)(4,15)(5,19,21,13)(6,14)(8,24)(9,23,18,28)(10,11)(12,17)(16,29,20,30)(22,26)(25,32,27,31)(33,35)(34,39)(36,43,47,60)(37,49)(38,45)(40,52,55,58)(41,53)(42,54)(44,51)(46,59,50,62)(48,61)(56,64,57,63);(1,4)(2,8)(3,12)(5,20)(6,17)(7,22)(9,27)(10,26)(11,24)(13,30)(14,15)(16,21)(18,25)(19,29)(23,32)(28,31)(40,55)(46,50)(52,58)(59,62);(1,3)(2,7,10,11)(4,15,17,12)(5,19)(6,14)(8,22)(9,23,18,28)(13,21)(16,30,20,29)(24,26)(25,32)(27,31)(33,36)(34,40)(35,43,45,60)(37,50)(38,47)(39,52,54,58)(41,57)(42,55)(44,59,49,62)(46,51)(48,56)(53,63,61,64);(1,5,6,21)(2,9)(3,13,14,19)(4,16)(7,23)(8,25,26,27)(10,18)(11,28)(12,29)(15,30)(17,20)(22,31,24,32)(33,37)(34,41)(35,44,45,49)(36,46)(38,51)(39,53,54,61)(40,56)(42,48)(43,59,60,62)(47,50)(52,63,58,64)(55,57)

Polycyclic presentation:

2^10|a^2=g,b^2=g,d^2=fg,e^2=fgi,b^a=bgi,c^a=cg,c^b=ci,d^a=dh,d^b=df,d^c=dfghj,e^a=efij,e^b=efhj,e^c=eg,e^d=efgij,f^a=fi,f^b=fj,f^c=fi,f^d=fj,f^e=fi,g^d=gj,g^e=gi,h^a=hj,h^b=hij,h^c=hi,h^d=hi,h^e=hij

Conjugacy classes & character table

58 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72341033, #72341091, #72341110
maximal normal subgroup of	#72341033, #72341091, #72341110

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