#72225544

order	2048
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₈ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	D₄×#322
commutator	C₂×C₈	C₂²×C₄×C₈
Frattini, cosocle	C₂×#346	C₂⁴
socle	C₂	D₄×#1313
prime radical	C₂×#135	#97
LCS	G ▷ C₂×C₈ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂×#346 ◁ G
LECS	G ▷ C₂×#346 ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1413 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 64 points:

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

Polycyclic presentation:

2^11|a^2=k,b^2=f,c^2=e,d^2=gjk,e^2=h,f^2=i,g^2=k,i^2=j,j^2=k,c^a=cg,c^b=cgijk,d^a=dg,d^b=dgk,e^a=ek,e^b=ejk,f^c=fj,f^e=fk,g^a=gk,g^b=gk,h^b=hk,i^c=ik

Conjugacy classes & character table

224 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72225554, #72227158, #72227179
maximal normal subgroup of	#72225554, #72227158, #72227179

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