#72227038

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄×C₈ · Schur multiplier (H₂) = C₂⁸×C₄

Aliases

No aliases for this group.

Subgroups

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

3927 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)(11,17)(12,26)(13,19)(14,20)(15,18)(16,21)(22,36)(23,31)(24,30)(25,32)(27,34)(28,33)(29,35)(37,40)(38,39);(2,5)(6,8)(9,11,14,23,15,24,27,38,16,25,13,29,12,22,28,37)(10,17,20,31,18,30,34,39,21,32,19,35,26,36,33,40);(9,12,16,15)(10,18,21,26)(11,22,25,24)(13,27,14,28)(17,30,32,36)(19,33,20,34)(23,37,29,38)(31,39,35,40);(1,2,3,6,4,5,7,8);(9,13,15,28,16,14,12,27)(10,19,18,33,21,20,26,34)(11,23,24,38,25,29,22,37)(17,31,30,39,32,35,36,40)

Polycyclic presentation:

2^10|b^2=f,c^2=j,d^2=g,e^2=h,f^2=h,g^2=i,h^2=j,c^a=cj,d^b=di,e^b=ej

Conjugacy classes & character table

352 conjugacy classes — view the full character table.

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