#72240801

order	2048
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#108082
commutator	C₂×#97	C₂×C₄²
Frattini, cosocle	C₂²×#405	C₂³
prime radical	C₂³×#135	D₄
LCS	G ▷ C₂×#97 ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂²×#405 ◁ G
LECS	G ▷ C₂²×#405 ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3635 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:

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

Polycyclic presentation:

2^11|a^2=hi,b^2=egk,c^2=f,d^2=i,e^2=j,g^2=k,j^2=k,b^a=be,c^b=cd,d^a=dh,d^b=dk,e^a=ej,e^b=ek,e^c=ehk,e^d=ek,f^b=fi,g^a=gk,g^b=gk,g^c=ghi,g^d=gk,h^b=hk,j^a=jk

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72307858, #72308088
maximal normal subgroup of	#72307858, #72308088

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