#72224964

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₈	#3184
commutator	C₂²×C₈	C₂³×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂⁴
socle	C₂	#210968
prime radical	#13309	C₄
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ C₂²×#346 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

896 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:

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

Polycyclic presentation:

2^10|a^2=e,b^2=ehj,c^2=g,e^2=h,g^2=i,h^2=j,i^2=j,b^a=bj,c^a=cg,c^b=ci,d^a=df,d^c=dhij,e^c=ej,f^c=fj,g^a=gi,g^b=gj,i^a=ij

Conjugacy classes & character table

112 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72224969, C₂×#72224964, #72343439, …
maximal normal subgroup of	#72224969, C₂×#72224964, #72343439, …
maximal quotient of	C₂×#72224964

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