#149340

order	512
nilpotency class	4
isoclinism	isoclinic class of #140874
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1982
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂	#11357
prime radical	C₂⁴	#137
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

469 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)(3,15,20,37)(4,21)(5,26)(6,30)(7,12)(8,14)(9,18,39,56)(10,23)(11,29)(13,16)(17,51,54,36)(19,46,34,50)(22,60,57,41)(24,47)(25,42)(27,38,31,40)(28,43)(32,44)(33,48)(35,59)(45,62)(49,52,63,53)(55,64,61,58);(1,3,16,19,8,20,7,34)(2,9,13,38,14,39,12,40)(4,22,24,17,25,57,35,54)(5,27,33,18,6,31,29,56)(10,41,43,36,45,60,44,51)(11,46,26,37,48,50,30,15)(21,58,47,49,42,64,59,63)(23,52,28,61,62,53,32,55);(1,4,16,24,8,25,7,35)(2,10,13,43,14,45,12,44)(3,17,19,57,20,54,34,22)(5,28,33,62,6,32,29,23)(9,36,38,60,39,51,40,41)(11,47,26,42,48,59,30,21)(15,49,46,64,37,63,50,58)(18,55,31,52,56,61,27,53)

Polycyclic presentation:

2^9|a^2=g,b^2=de,c^2=de,d^2=h,e^2=h,f^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ehi,h^a=hi,c^b=cf,d^b=dhi,e^b=eh,f^b=fi,g^b=gi,h^b=hi,d^c=dh,e^c=ehi,f^c=fi,h^c=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 56536);