#72322762

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄² · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator	C₂×C₄	C₂³×C₄²
Frattini, cosocle	C₄×C₈	C₂⁵
socle	C₂²	C₂³×#97
prime radical	C₂²×#41	C₄²
LCS	G ▷ C₂×C₄ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

3079 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^10|a^2=g,b^2=f,d^2=h,f^2=h,g^2=i,i^2=j,b^a=bi,c^b=ch,d^a=dj,d^b=dh,e^d=ej,f^a=fj,g^b=gj

Conjugacy classes & character table

232 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72322814, #72322990
maximal normal subgroup of	#72322814, #72322990

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