#72337397

order	2048
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂¹⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, socle, cosocle	C₂⁶	C₂⁵
commutator	C₂⁵	C₂⁴×C₄
prime radical	C₂⁶×D₄	C₂²
LCS	G ▷ C₂⁵ ▷ C₁
UCS	C₁ ◁ C₂⁶ ◁ G
LECS	G ▷ C₂⁶ ▷ C₁

826144 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 5 generators on 28 points:

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

Polycyclic presentation:

2^11|a^2=h,b^2=f,b^a=bhk,c^b=cg,d^b=di,d^c=dk,e^b=ehij

Conjugacy classes & character table

416 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, 1, F.8);;
gap> SetPower(coll, 2, F.6);;
gap> SetConjugate(coll, 2, 1, F.2*F.8*F.11);;
gap> SetConjugate(coll, 3, 2, F.3*F.7);;
gap> SetConjugate(coll, 4, 2, F.4*F.9);;
gap> SetConjugate(coll, 4, 3, F.4*F.11);;
gap> SetConjugate(coll, 5, 2, F.5*F.8*F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);