#148446

order	512
nilpotency class	3
isoclinism	isoclinic class of #2048
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₈	#394
commutator	C₂²×C₄	C₂²×C₈
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂	#11147
prime radical	C₂²×#41	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

637 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=d,d^2=i,e^2=i,g^2=h,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=ei,c^b=cf,d^b=di,f^c=fi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 55642);