#72320175

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	#442
commutator	C₂×C₈	C₂²×C₄²
Frattini, cosocle	C₂³×C₈	C₂⁴
socle	C₂³	C₂×#442
prime radical	C₂⁵	AGL(1,ℤ/8)
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂⁴×C₄ ◁ C₂³×#97 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

3041 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:

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

Polycyclic presentation:

2^10|a^2=e,b^2=f,c^2=efghij,g^2=i,h^2=i,i^2=j,b^a=bg,c^a=cfj,c^b=cfi,d^a=dj,d^b=dj,e^b=ej,g^a=gi,g^b=gij,g^c=gj,h^a=hi,h^b=hij,h^c=hj,i^a=ij,i^b=ij

Conjugacy classes & character table

136 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72320214, #72320242, #72320721
maximal normal subgroup of	#72320214, #72320242, #72320721

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