#72280652

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	#25834
commutator, Frattini, cosocle	C₂³×C₈	C₂⁴
socle	C₂	#403380
prime radical	C₂²×#135	C₂³
LCS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂×#2961 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

3743 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:

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

Polycyclic presentation:

2^10|a^2=g,b^2=g,c^2=g,d^2=fhij,e^2=j,f^2=j,h^2=i,i^2=j,c^a=cg,c^b=cg,d^a=df,d^b=dfhij,d^c=de,f^b=fj,f^c=fj,g^d=gj,h^a=hj,h^b=hi,i^b=ij

Conjugacy classes & character table

88 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72307810
maximal normal subgroup of	#72307810

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