#72237294

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	C₂×#1804
commutator	C₂²×C₈	C₂³×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂⁴
prime radical	C₂³×D₈	C₂×C₄
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ C₂×#1726 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

2277 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:

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

Polycyclic presentation:

2^10|a^2=f,b^2=f,c^2=e,d^2=j,e^2=h,g^2=i,h^2=j,i^2=j,c^a=cg,c^b=cfij,d^a=dj,d^c=dhij,e^a=ei,f^c=fj,g^a=gi,g^b=gj,h^a=hj,i^a=ij

Conjugacy classes & character table

124 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72238783, #72238791, #72239689, …
maximal normal subgroup of	#72238783, #72238791, #72239689, …

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