#72287445

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#10889
commutator	C₂×M₁₆	C₂×C₄²
Frattini, cosocle	C₄×#131	C₂³
socle	C₂	#109915
prime radical	C₂×#41	C₄×D₄
LCS	G ▷ C₂×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂×C₈ ◁ C₄×#131 ◁ G
LECS	G ▷ C₄×#131 ▷ C₂×C₄ ▷ C₂ ▷ C₁

517 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:

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

Polycyclic presentation:

2^10|a^2=d,b^2=dfi,c^2=dfgi,d^2=hij,e^2=ij,f^2=ij,g^2=i,i^2=j,b^a=be,c^a=cfj,c^b=ce,d^b=dh,e^a=ehi,e^c=eij,e^d=ej,f^a=fi,f^b=fh,f^c=fij,f^e=fj,g^a=gij,g^b=gj,g^c=gij,h^a=hj,h^c=hj,i^a=ij,i^c=ij

Conjugacy classes & character table

76 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72287460, #72287462, #72343157, …
maximal normal subgroup of	#72287460, #72287462, #72343157, …

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