#72309806

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	D₄×#106
commutator	C₄×C₈	C₂³×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂⁴
prime radical	C₂²×D₄	C₂×SD₁₆
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₈×SD₁₆ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

849 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)(11,16)(12,22,23,35)(13,26)(14,29)(15,17)(18,24,31,36)(19,27)(20,33,25,38)(21,32)(28,39)(30,34,40,37);(2,6)(3,7)(5,8)(9,11,15,14)(10,16,17,29)(12,20,23,25)(13,21,28,27)(18,30,31,40)(19,26,32,39)(22,33,35,38)(24,34,36,37);(1,2,4,5)(3,8,7,6)(10,17)(16,29)(18,31)(19,32)(22,35)(26,39)(30,40)(33,38);(9,12,13,24,14,25,27,37,15,23,28,36,11,20,21,34)(10,18,19,22,16,30,26,33,17,31,32,35,29,40,39,38)

Polycyclic presentation:

2^10|a^2=e,b^2=j,c^2=h,d^2=g,f^2=h,g^2=i,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	#72309828, #72309983, #72310110
maximal normal subgroup of	#72309828, #72309983, #72310110

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.8);;
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, 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);