#72331202

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

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 19085 conjugacy classes of subgroups; only the 5344 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:

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

Polycyclic presentation:

2^11|f^2=ij,b^a=bk,c^b=cg,d^a=dj,d^b=di,e^d=eg,f^d=fij,f^e=fgh

Conjugacy classes & character table

248 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72331654
maximal normal subgroup of	#72331654

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