#72333263

order	2048
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹⁴×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 33223 conjugacy classes of subgroups; only the 5024 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:

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

Polycyclic presentation:

2^11|a^2=j,c^2=j,b^a=bk,c^b=cg,d^a=dh,d^c=dh,e^a=ej,e^c=ej,e^d=eh,f^a=fj,f^b=fg,f^d=fhi,f^e=fg

Conjugacy classes & character table

212 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72333354
maximal normal subgroup of	#72333354

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