#72323737

order	2048
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁵	C₂⁶
LCS	G ▷ C₂⁵ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₁

At least 13327 conjugacy classes of subgroups; only the 5856 normal subgroups are shown.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 28 points:

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

Polycyclic presentation:

2^11|c^2=hk,d^2=ghi,e^2=ik,f^2=hk,c^a=cg,c^b=cgh,d^b=dghi,d^c=dik,e^b=eik,e^c=eghi,e^d=eghk,f^d=fj,f^e=fhjk

Conjugacy classes & character table

224 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72323772
maximal normal subgroup of	#72323772

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