#72341811

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#24260
commutator	C₂×#41	C₂³×C₄
Frattini, cosocle	C₂²×#41	C₂⁴
socle	C₂	#360155
prime radical	C₂×#3018	C₂²
LCS	G ▷ C₂×#41 ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂²×#41 ▷ C₂² ▷ C₂ ▷ C₁

3489 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 64 points:

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

Polycyclic presentation:

2^10|a^2=e,c^2=f,d^2=j,e^2=j,f^2=j,g^2=j,c^a=cf,d^a=dh,d^b=dg,f^a=fj,g^a=gi,g^b=gj,g^d=gj,h^b=hij,h^g=hj,i^d=ij

Conjugacy classes & character table

100 conjugacy classes — view the full character table.

Relationships

maximal subgroup of

#72341825

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