#6468122

order	512
nilpotency class	3
isoclinism	isoclinic class of #24986
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₈	#555
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂	#34943
prime radical	#28250	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1060 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 5 generators on 64 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=gh,d^2=h,e^2=f,f^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=dh,h^b=hi,d^c=di

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6375318);