#72317524

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#9497
commutator	C₂²×#41	C₂²×C₄
Frattini, cosocle	C₂³×#41	C₂³
socle	C₂	#115133
prime radical	#6437463	C₂
LCS	G ▷ C₂²×#41 ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂³×#41 ◁ G
LECS	G ▷ C₂³×#41 ▷ C₂³ ▷ C₂ ▷ C₁

3878 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 64 points:

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

Polycyclic presentation:

2^10|a^2=e,b^2=ij,c^2=h,e^2=j,b^a=bf,c^a=cd,c^b=cdefgj,d^a=dh,d^b=dj,e^b=ei,e^c=eh,f^a=fi,f^c=fhj,f^d=fj,g^a=gi,g^b=gij,g^c=gj,g^d=gj,h^b=hj,i^c=ij

Conjugacy classes & character table

70 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72317535, #72317554, #72317758
maximal normal subgroup of	#72317535, #72317554, #72317758

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