#72225411

order	2048
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	C₂×D₈²
commutator	C₄×C₈	C₂⁴×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂⁵
socle	C₂	C₄×D₈²
prime radical	#72225394	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₈ ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

5398 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 64 points:

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

Polycyclic presentation:

2^11|a^2=k,c^2=fgij,d^2=gj,e^2=h,f^2=ik,g^2=jk,i^2=k,j^2=k,c^a=cf,c^b=cg,d^b=dg,e^a=ei,e^d=ek,f^a=fi,g^b=gj,h^a=hk,i^a=ik,j^b=jk

Conjugacy classes & character table

212 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72225512, #72226910, #72226961, …
maximal normal subgroup of	#72225512, #72226910, #72226961, …

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