#72227241

order	2048
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₄×C₈ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#93073
commutator	C₈²	C₄×C₈
Frattini, cosocle	C₈×#346	C₂²
socle	C₂	#72216557
prime radical	C₂³	#7270
LCS	G ▷ C₈² ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₄²×C₈ ◁ C₈×#346 ◁ G
LECS	G ▷ C₈×#346 ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

398 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 2 generators on 64 points:

(1,2,4,12,6,14,28,43,27,17,34,46,7,15,29,44,10,18,31,47,8,13,5,32,9,16,30,45,33,49,57,59)(3,11,19,37,20,39,52,60,24,51,21,56,36,48,55,64,25,42,54,62,35,50,58,63,23,41,53,61,22,40,26,38);(1,3,5,21,33,20,31,55,9,24,29,54,8,36,34,58,10,25,28,53,7,35,4,26,27,23,57,19,6,22,30,52)(2,11,12,38,13,48,32,56,17,41,46,63,49,39,59,37,18,42,47,64,14,40,43,61,16,51,45,60,15,50,44,62)

Polycyclic presentation:

2^11|a^2=c,b^2=e,c^2=f,d^2=ghijk,e^2=hik,f^2=ik,g^2=i,h^2=jk,i^2=k,j^2=k,b^a=bd,c^b=cg,d^a=dh,d^b=dgij,e^a=ehik,e^c=ek,f^b=fi,g^b=gi,h^a=hj,h^b=hk,i^b=ik,j^a=jk

Conjugacy classes & character table

140 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72227242, #72227250, #72227260, …
maximal normal subgroup of	#72227242, #72227250, #72227260, …

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