#72339938

order	2048
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹⁷

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 6902 conjugacy classes of subgroups; only the 7816 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:

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

Polycyclic presentation:

2^11|c^2=k,e^2=hijk,b^a=bg,c^b=cj,d^b=di,d^c=dk,e^a=eh,e^b=egijk,e^c=ej,e^d=ei,f^b=fghk,f^e=fghk

Conjugacy classes & character table

272 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72339945, #72339997
maximal normal subgroup of	#72339945, #72339997

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