#72303686

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	C₂⁴×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

12517 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:

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

Polycyclic presentation:

2^10|a^2=g,d^2=j,e^2=j,c^b=cgh,d^c=di,e^b=eh,e^c=ei,e^d=ej,f^a=fhi

Conjugacy classes & character table

154 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72303693
maximal normal subgroup of	#72303693

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