#72340106

order	2048
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂¹⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, socle, cosocle	C₂⁶	C₂⁵
commutator	C₂⁵	C₂⁴×C₄
prime radical	C₂⁵×#121	C₂
LCS	G ▷ C₂⁵ ▷ C₁
UCS	C₁ ◁ C₂⁶ ◁ G
LECS	G ▷ C₂⁶ ▷ C₁

832760 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^11|b^2=f,c^2=k,d^2=fi,b^a=bg,c^b=cj,d^a=dh,d^b=dghi,d^c=djk,e^b=eghk,e^d=eghk

Conjugacy classes & character table

320 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", "k");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 2, F.6);;
gap> SetPower(coll, 3, F.11);;
gap> SetPower(coll, 4, F.6*F.9);;
gap> SetConjugate(coll, 2, 1, F.2*F.7);;
gap> SetConjugate(coll, 3, 2, F.3*F.10);;
gap> SetConjugate(coll, 4, 1, F.4*F.8);;
gap> SetConjugate(coll, 4, 2, F.4*F.7*F.8*F.9);;
gap> SetConjugate(coll, 4, 3, F.4*F.10*F.11);;
gap> SetConjugate(coll, 5, 2, F.5*F.7*F.8*F.11);;
gap> SetConjugate(coll, 5, 4, F.5*F.7*F.8*F.11);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);