#72267499

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	C₂²×#120	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

16289 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 32 points:

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

Polycyclic presentation:

2^10|c^2=gh,d^2=gi,e^2=j,f^2=j,c^a=cg,c^b=ci,d^a=di,d^b=dhi,d^c=di,e^d=eij,f^c=fij,f^d=fij,f^e=fj

Conjugacy classes & character table

136 conjugacy classes — view the full character table.

Relationships

maximal quotient of

#72267510

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