#72309533

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×D₄×D₈
commutator, Frattini, cosocle	C₄×C₈	C₂⁵
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₈×SD₁₆ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

2706 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

100 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72309550, #72309828
maximal normal subgroup of	#72309550, #72309828

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