#72227057

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

3843 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)(11,16)(12,24)(13,18)(14,17)(15,19)(20,28)(21,34)(22,29)(23,30)(25,37)(26,32)(27,31)(33,38)(35,40)(36,39);(1,2,4,7,5,8,3,6)(11,20,22,21)(13,26)(16,28,29,34)(18,32)(23,35,36,33)(25,27)(30,40,39,38)(31,37);(9,11,13,23,14,20,27,33,15,22,26,36,12,21,25,35)(10,16,18,30,17,28,31,38,19,29,32,39,24,34,37,40);(9,12,15,14)(10,17,19,24)(11,21,22,20)(13,25,26,27)(16,28,29,34)(18,31,32,37)(23,35,36,33)(30,38,39,40);(1,3,5,4)(2,6,8,7)

Polycyclic presentation:

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

Conjugacy classes & character table

328 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.7);;
gap> SetPower(coll, 4, F.10);;
gap> SetPower(coll, 5, F.9);;
gap> SetPower(coll, 6, F.9);;
gap> SetPower(coll, 7, F.8);;
gap> SetPower(coll, 8, F.10);;
gap> SetConjugate(coll, 3, 2, F.3*F.8);;
gap> SetConjugate(coll, 4, 1, F.4*F.10);;
gap> SetConjugate(coll, 6, 3, F.6*F.10);;
gap> SetConjugate(coll, 7, 2, F.7*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);