#72292964

order	1024
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#2127
commutator	C₂×C₄²	C₂³×C₄
Frattini, cosocle	C₂×#124	C₂⁴
prime radical	C₂×#3151	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ #34070 ◁ G
LECS	G ▷ C₂×#124 ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁

1832 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:

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

Polycyclic presentation:

2^10|b^2=e,d^2=gij,e^2=j,f^2=ij,h^2=j,b^a=bf,c^b=cg,d^a=di,e^a=eh,e^c=ej,f^a=fij,f^b=fhij,f^d=fj,f^e=fi,g^b=gj,h^a=hj,h^b=hi,h^e=hj,i^b=ij

Conjugacy classes & character table

70 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72292966, #72292985, #72293082, …
maximal normal subgroup of	#72292966, #72292985, #72293082, …

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