#72265049

order	2048
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#114110
commutator	C₂²×C₄×C₈	C₂²×C₄
Frattini, cosocle	C₂²×#405	C₂³
prime radical	C₂²×#469	D₄
LCS	G ▷ C₂²×C₄×C₈ ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂²×#405 ◁ G
LECS	G ▷ C₂²×#405 ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3349 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 48 points:

(1,2)(3,9,11,15)(4,7)(5,8)(6,10,13,16)(12,14)(17,18,20,23,21,25,31,37,22,26,32,38,34,43,46,42)(19,24,28,35,30,40,45,48,27,41,33,36,29,39,44,47);(6,13)(9,15)(18,23,26,38)(19,27)(20,31)(21,34)(24,35,41,36)(25,42,43,37)(28,44)(32,46)(33,45)(39,48,40,47);(1,3)(2,6,8,13)(4,10)(5,11)(7,9,14,15)(12,16)(18,24)(20,32)(23,35)(25,40)(26,41)(28,33)(31,46)(36,38)(37,48)(39,43)(42,47)(44,45)

Polycyclic presentation:

2^11|a^2=e,b^2=hk,c^2=f,d^2=i,e^2=j,g^2=k,j^2=k,b^a=begk,c^a=cd,c^b=cf,d^a=dh,d^b=dk,e^b=ej,e^c=ehi,e^d=ek,f^a=fi,g^b=gk,g^c=ghk,g^d=gk,h^a=hk,j^b=jk

Conjugacy classes & character table

110 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72265173, #72307393
maximal normal subgroup of	#72265173, #72307393

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