#72225421

order	2048
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#225516
commutator	C₄×C₈	C₂³×C₈
Frattini, cosocle	C₄²×C₈	C₂⁴
socle	C₂	#72216394
prime radical	C₂×#508	C₄⋊C₄
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₈ ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₄²×C₈ ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1390 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:

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

Polycyclic presentation:

2^11|a^2=k,b^2=e,c^2=fgk,d^2=gi,e^2=h,f^2=ik,g^2=jk,i^2=k,j^2=k,b^a=bf,c^a=cf,c^b=cg,d^b=dgik,e^a=ei,e^d=ek,f^a=fi,f^b=fk,g^a=gk,g^b=gj,h^a=hk,i^a=ik,j^b=jk

Conjugacy classes & character table

188 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72225512, #72226993, #72227037, …
maximal normal subgroup of	#72225512, #72226993, #72227037, …

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*F.7*F.11);;
gap> SetPower(coll, 4, F.7*F.9);;
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, 2, 1, F.2*F.6);;
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*F.9*F.11);;
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, 6, 2, F.6*F.11);;
gap> SetConjugate(coll, 7, 1, F.7*F.11);;
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);