#72225538

order	2048
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₈ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#96688
commutator	C₄×C₈	C₂×C₄×C₈
Frattini, cosocle	C₄×#346	C₂³
socle	C₂	#72225537
prime radical	C₂×#135	#97
LCS	G ▷ C₄×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂×#346 ◁ C₄×#1398 ◁ G
LECS	G ▷ C₄×#346 ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

782 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 64 points:

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

Polycyclic presentation:

2^11|a^2=k,b^2=e,c^2=d,d^2=g,e^2=h,f^2=ik,h^2=j,i^2=k,j^2=k,c^a=cf,c^b=cfhjk,d^a=di,d^b=dij,e^c=ej,e^d=ek,f^a=fi,f^b=fi,f^c=fk,g^a=gk,h^c=hk,i^a=ik,i^b=ik

Conjugacy classes & character table

176 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72225554, #72227138, #72227175
maximal normal subgroup of	#72225554, #72227138, #72227175

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