#72337290

order	2048
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶×C₄ · Schur multiplier (H₂) = C₂²⁰

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, socle, cosocle	C₂⁴	C₂⁷
commutator	C₂³	C₂⁶×C₄
prime radical	C₂⁴×D₄²	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

At least 7347 conjugacy classes of subgroups; only the 83728 normal subgroups are shown.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 7 generators on 24 points:

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

Polycyclic presentation:

2^11|a^2=j,g^2=h,b^a=bj,d^c=dk,g^a=gjk,g^b=gi,g^c=gj,g^d=gi,g^e=gi,g^f=gj

Conjugacy classes & character table

392 conjugacy classes — view the full character table.

Relationships

maximal quotient of

#72337321, #72337359

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