#72232676

order	2048
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁷ · Schur multiplier (H₂) = C₂¹⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁷
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

At least 33460 conjugacy classes of subgroups; only the 29836 normal subgroups are shown.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 7 generators on 32 points:

(11,15)(14,16)(19,23)(22,24)(26,30)(28,32);(10,13)(14,16)(18,21)(22,24)(26,30)(27,31);(2,5)(6,8)(10,13)(14,16)(17,18)(19,22)(20,21)(23,24)(25,26,29,30)(27,32,31,28);(2,5)(3,7)(10,13)(11,15)(19,23)(22,24);(2,5)(6,8)(10,13)(11,15)(19,23)(22,24)(27,31)(28,32);(1,2)(3,6)(4,5)(7,8)(9,10)(11,14)(12,13)(15,16)(17,18)(19,22)(20,21)(23,24)(25,27)(26,28)(29,31)(30,32);(1,3)(2,6)(4,7)(5,8)(9,11)(10,14)(12,15)(13,16)(17,19)(18,22)(20,23)(21,24)(25,28)(26,27)(29,32)(30,31)

Polycyclic presentation:

2^11|c^2=i,c^a=ci,c^b=cij,f^b=fh,f^c=fik,f^d=fk,f^e=fik,g^a=gh,g^d=gjk,g^e=gh

Conjugacy classes & character table

218 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72233213, #72233347
maximal normal subgroup of	#72233213, #72233347
automorphism group of	#1510, #1511

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