#72285282

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, socle	C₂²	#32898
commutator, Frattini, cosocle	C₂²×C₂²⋊C₄	C₂⁴
LCS	G ▷ C₂²×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂²×#120 ◁ G
LECS	G ▷ C₂²×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

4371 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 40 points:

(2,5)(3,8)(13,26)(17,32)(19,33)(20,23)(21,27)(25,38)(28,39)(31,40);(1,2)(3,6)(4,5)(7,8)(9,10)(11,19,22,33)(12,16)(13,17)(14,18)(15,21,29,27)(20,34,23,37)(24,30)(25,31)(26,32)(28,36,39,35)(38,40);(2,6)(3,8)(5,7)(9,11)(10,15,18,29)(12,23)(13,27,26,21)(14,22)(16,28,30,39)(17,33)(19,32)(20,24)(25,35,38,36)(31,34)(37,40);(9,12)(10,16)(11,20)(13,25)(14,24)(15,28)(17,31)(18,30)(19,34)(21,36)(22,23)(26,38)(27,35)(29,39)(32,40)(33,37)

Polycyclic presentation:

2^10|b^2=i,c^2=g,f^2=hj,b^a=bhi,c^a=ce,c^b=cf,d^a=di,d^c=dgj,e^b=eh,e^d=ej,f^a=fhj,f^b=fh,f^c=fhi,f^d=fi,g^b=gij,g^f=gj,i^c=ij

Conjugacy classes & character table

61 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72285301, #72341243
maximal normal subgroup of	#72285301, #72341243

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