#72225430

order	2048
nilpotency class	6
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#72225429
commutator, Frattini, cosocle	C₄×C₃₂	C₂⁴
LCS	G ▷ C₄×C₃₂ ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ C₂×C₈×C₁₆ ◁ C₈×M₆₄ ◁ G
LECS	G ▷ C₄×C₃₂ ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

789 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:

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

Polycyclic presentation:

2^11|a^2=k,d^2=fh,e^2=gi,f^2=h,g^2=ij,i^2=jk,j^2=k,b^a=be,c^a=ck,c^b=ch,d^a=di,d^b=df,e^a=eg,e^b=eg,e^c=ek,f^a=fj,f^b=fh,g^a=gi,g^b=gi,h^a=hk,i^a=ij,i^b=ij,j^a=jk,j^b=jk

Conjugacy classes & character table

134 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72225451, #72225510, #72225515
maximal normal subgroup of	#72225451, #72225510, #72225515

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