#72283161

order	2048
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁷ · Schur multiplier (H₂) = C₂²⁰

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

365476 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 7 generators on 28 points:

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

Polycyclic presentation:

2^11|c^2=h,d^2=k,c^b=ck,d^a=dk,d^c=di,e^a=eh,e^b=eij,e^d=eh,f^b=fk,f^c=fh,g^b=gj

Conjugacy classes & character table

248 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72284919
maximal normal subgroup of	#72284919
maximal quotient of	#72292028

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