#72309648

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	#22670
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 ◁ #8283 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

830 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:

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

Polycyclic presentation:

2^10|a^2=e,b^2=fh,c^2=eh,d^2=g,f^2=h,g^2=i,i^2=j,b^a=bf,c^a=cf,c^b=cfj,d^a=dg,d^b=dj,d^c=di,e^d=ej,f^a=fh,f^c=fh,g^a=gi,g^c=gj,i^a=ij

Conjugacy classes & character table

82 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72309657, #72309828
maximal normal subgroup of	#72309657, #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.5);;
gap> SetPower(coll, 2, F.6*F.8);;
gap> SetPower(coll, 3, F.5*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, 2, 1, F.2*F.6);;
gap> SetConjugate(coll, 3, 1, F.3*F.6);;
gap> SetConjugate(coll, 3, 2, F.3*F.6*F.10);;
gap> SetConjugate(coll, 4, 1, F.4*F.7);;
gap> SetConjugate(coll, 4, 2, F.4*F.10);;
gap> SetConjugate(coll, 4, 3, F.4*F.9);;
gap> SetConjugate(coll, 5, 4, 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, 3, F.7*F.10);;
gap> SetConjugate(coll, 9, 1, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);