#72225366

order	2048
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₈ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₁₆	#2158
commutator	C₂×C₁₆	C₂×C₄×C₈
Frattini, cosocle	C₂×C₈×C₁₆	C₂³
socle	C₂	#72225365
prime radical	C₂³	#12046
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₁₆ ◁ C₂×M₆₄ ◁ C₄×M₆₄ ◁ C₈×M₆₄ ◁ G
LECS	G ▷ C₂×C₈×C₁₆ ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

451 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 64 points:

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

Polycyclic presentation:

2^11|a^2=d,b^2=e,c^2=fhij,d^2=k,e^2=g,f^2=h,g^2=i,h^2=j,j^2=k,b^a=bf,c^a=cfijk,d^c=dk,e^a=eh,f^a=fhjk,g^a=gj,h^a=hjk,i^a=ik,j^a=jk

Conjugacy classes & character table

368 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72225393, #72225481, #72225495, …
maximal normal subgroup of	#72225393, #72225481, #72225495, …

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