#72237628

order	2048
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#2403
commutator, Frattini, cosocle	C₂⁵	C₂⁶
LCS	G ▷ C₂⁵ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁷ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

At least 10467 conjugacy classes of subgroups; only the 5164 normal subgroups are shown.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 6 generators on 24 points:

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

Polycyclic presentation:

2^11|e^2=ik,b^a=bg,c^a=cj,c^b=ch,d^b=di,d^c=dk,e^a=eik,e^b=eik,e^c=ej,e^d=ei,f^b=fi,g^c=gk,h^e=hk,j^b=jk

Conjugacy classes & character table

188 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72237631, #72238348, #72238353, …
maximal normal subgroup of	#72237631, #72238348, #72238353, …

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