#72307196

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#25671
commutator, Frattini, cosocle	C₂³×C₈	C₂⁴
socle	C₂	#404277
prime radical	#10575347	C₂
LCS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂×#2875 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

4116 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:

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

Polycyclic presentation:

2^10|a^2=j,c^2=e,d^2=f,e^2=i,g^2=ij,h^2=j,i^2=j,b^a=bf,c^a=cg,c^b=cghj,d^c=deghij,e^a=ei,e^b=eij,e^d=ej,f^c=fj,g^a=gi,g^b=gij,g^c=gj,i^a=ij,i^b=ij

Conjugacy classes & character table

94 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72307305
maximal normal subgroup of	#72307305

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