#72225383

order	2048
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₈ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₈	C₄²×D₈
commutator	C₂×C₈	C₂²×C₄×C₈
Frattini, cosocle	C₂×C₄×C₁₆	C₂⁴
socle	C₂	D₈×#346
prime radical	C₂×#508	C₂×C₈
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂×#1398 ◁ C₄×#1398 ◁ G
LECS	G ▷ C₂×C₄×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1386 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:

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

Polycyclic presentation:

2^11|a^2=k,b^2=e,c^2=f,d^2=gj,f^2=h,g^2=i,h^2=j,i^2=k,j^2=k,b^a=bi,c^a=cg,c^b=cj,d^a=dgi,d^b=dk,e^a=ek,e^c=ek,f^a=fi,f^b=fk,g^a=gik,h^a=hk,i^a=ik

Conjugacy classes & character table

296 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72225393, #72225554, #72226910, …
maximal normal subgroup of	#72225393, #72225554, #72226910, …

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