#72267729

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹¹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁶
prime radical	#15257	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

At least 6942 conjugacy classes of subgroups; only the 3085 normal subgroups are shown.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^10|d^2=gi,e^2=j,f^2=j,c^a=ch,c^b=cg,d^a=dhi,d^b=dghj,d^c=dij,e^d=eij,f^c=fij,f^e=fj

Conjugacy classes & character table

130 conjugacy classes — view the full character table.

Relationships

maximal quotient of

#72267741

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