#72309783

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

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	#490472	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #8202 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

987 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^10|a^2=e,b^2=j,c^2=ehj,d^2=g,f^2=h,g^2=i,i^2=j,c^a=chj,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	#72309802, #72309828, #72310167
maximal normal subgroup of	#72309802, #72309828, #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.8*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*F.10);;
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);