#72309455

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	#23001
commutator	C₄×C₈	C₂³×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂⁴
prime radical	#33582	C₂²
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #8283 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

935 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^10|a^2=e,b^2=f,c^2=eh,d^2=egi,f^2=h,g^2=i,i^2=j,c^a=cj,c^b=cfhj,d^a=dg,d^c=di,e^d=ej,f^c=fh,g^a=gi,g^c=gj,g^d=gj,i^a=ij

Conjugacy classes & character table

94 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72309470, #72309705
maximal normal subgroup of	#72309470, #72309705

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