#72309643

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	#21503
commutator	C₄×C₈	C₂³×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂⁴
prime radical	C₂×#135	C₂×D₄
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #7846 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

958 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)(3,6)(7,8)(9,10)(11,19,22,35)(12,25)(13,27)(14,16)(15,29,31,36)(17,20)(18,28)(21,32,24,34)(23,30,38,40)(26,39)(33,37);(1,2,6,8,4,7,3,5);(1,2)(3,8)(4,7)(5,6)(9,11,20,36,28,21,12,23,14,22,37,29,13,24,26,38)(10,15,25,35,27,40,17,32,16,31,39,19,18,30,33,34);(2,7)(5,8)(10,16)(11,21,22,24)(12,26)(15,30,31,40)(18,27)(19,34,35,32)(20,37)(23,36,38,29)

Polycyclic presentation:

2^10|a^2=e,b^2=fh,c^2=gi,d^2=e,f^2=h,g^2=i,i^2=j,b^a=bf,c^a=cfgij,c^b=cfh,d^a=dj,d^b=dh,d^c=dhi,e^c=ej,f^a=fh,f^c=fh,g^a=gi,g^d=gj,i^a=ij

Conjugacy classes & character table

82 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72309657, #72309851
maximal normal subgroup of	#72309657, #72309851

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