#72225447

order	2048
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#220728
commutator	C₄×C₈	C₂³×C₈
Frattini, cosocle	C₄²×C₈	C₂⁴
socle	C₂	#72216473
prime radical	#7623661	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

2733 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

164 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72225512, #72227138, #72227168
maximal normal subgroup of	#72225512, #72227138, #72227168

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.5);;
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, 4, 2, F.4*F.7);;
gap> SetConjugate(coll, 4, 3, F.4*F.10);;
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, 3, F.6*F.11);;
gap> SetConjugate(coll, 7, 2, F.7*F.10);;
gap> SetConjugate(coll, 7, 3, F.7*F.11);;
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);