#72227396

order	2048
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#7555
commutator	C₂⁶	C₂×C₄²
Frattini, cosocle	C₂⁸	C₂³
prime radical	C₂⁹	C₂²
LCS	G ▷ C₂⁶ ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁸ ◁ G
LECS	G ▷ C₂⁸ ▷ C₂³ ▷ C₁

2098628 conjugacy classes of subgroups; only the 1056 normal subgroups are shown.

normal subgroups only

This group has 2098628 conjugacy classes of subgroups — too many to draw. Only its 1056 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 32

A minimal faithful permutation representation — 3 generators on 24 points:

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

Polycyclic presentation:

2^11|b^2=g,c^2=f,b^a=be,c^a=cd,c^b=ch,d^b=di,e^c=ei,f^b=fj,g^c=gk,h^b=hk,h^c=hj

Conjugacy classes & character table

200 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72227398
maximal normal subgroup of	#72227398
maximal quotient of	#72267718, #72268055

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