#495104

order	512
nilpotency class	3
isoclinism	isoclinic class of #3068
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₈	C₂×#121
commutator	C₂×C₄	C₂³×C₈
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	C₂×#1954
prime radical	C₂×#575	C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1121 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,c^2=f,e^2=i,f^2=i,g^2=h,h^2=i,b^a=be,c^a=cf,f^a=fi,e^b=ei,g^b=gi,d^c=di

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 402300);