#72227194

order	2048
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₈ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	C₄×#106
commutator	C₄×C₈	C₂×C₄×C₈
Frattini, cosocle	C₂×C₈×C₁₆	C₂³
socle	C₂²	#129086
prime radical	C₂³	#12013
LCS	G ▷ C₄×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂×#1311 ◁ C₈×#346 ◁ G
LECS	G ▷ C₂×C₈×C₁₆ ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

538 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^11|a^2=d,b^2=eij,c^2=f,d^2=g,e^2=h,f^2=ij,g^2=i,h^2=j,i^2=k,b^a=be,c^a=ci,d^b=dj,d^c=dk,e^a=eh,f^a=fk,h^a=hj

Conjugacy classes & character table

272 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72227217, #72227255, #72227284
maximal normal subgroup of	#72227217, #72227255, #72227284
maximal quotient of	#72227195

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