#72309975

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#23029
commutator	C₄×C₈	C₂³×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂⁴
prime radical	C₂²×D₄	AGL(1,ℤ/8)
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #8202 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

832 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^10|a^2=e,b^2=j,c^2=ej,d^2=g,f^2=h,g^2=i,i^2=j,c^a=ch,c^b=cf,d^a=dg,d^c=di,e^d=ej,f^b=fh,f^c=fh,g^a=gi,g^c=gj,i^a=ij

Conjugacy classes & character table

73 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72309983, #72310021, #72310167
maximal normal subgroup of	#72309983, #72310021, #72310167

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