#72240002

order	2048
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#10995
commutator	C₂²×C₄⋊C₄	C₂²×C₈
Frattini, cosocle	C₂²×#405	C₂³
socle	C₂²	#111210
prime radical	#609717	C₄
LCS	G ▷ C₂²×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₈ ◁ C₂²×#405 ◁ G
LECS	G ▷ C₂²×#405 ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3909 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)(6,13)(7,14)(10,16)(18,24,28,29)(20,33)(25,38,26,37)(27,43,44,39)(30,41)(34,35)(40,47,42,48)(45,46);(1,2)(3,9,11,15)(4,7)(5,8)(6,10,13,16)(12,14)(17,18,20,27,23,29,35,43,22,28,33,44,21,24,34,39)(19,25,30,40,31,38,45,48,32,26,41,42,36,37,46,47);(1,3)(2,6,8,13)(4,10)(5,11)(7,9,14,15)(12,16)(18,25,28,26)(24,37,29,38)(27,40,44,42)(39,47,43,48)

Polycyclic presentation:

2^11|a^2=hik,b^2=e,c^2=f,d^2=i,e^2=j,g^2=k,j^2=k,b^a=bg,c^a=cf,c^b=cd,d^a=dh,d^b=dk,e^a=ek,e^c=eik,f^b=fi,g^c=ghik,g^d=gk,h^b=hk

Conjugacy classes & character table

140 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72240705, #72265173, #72265475, …
maximal normal subgroup of	#72240705, #72265173, #72265475, …

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