#6468120

order	512
nilpotency class	3
isoclinism	isoclinic class of #6447056
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	C₂×#555
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂	#34943
prime radical	#28249	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₈ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1056 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)(5,12,31,46,32,47,6,13)(18,35,24,40,51,61,56,60)(19,45,30,37,20,49,53,36)(25,42,26,54,57,48,33,41)(50,59,55,63,52,62,58,64);(1,3,7,21,9,22,8,34)(2,10,15,38,16,39,14,17)(4,24,27,51,29,56,28,18)(5,30,6,20,32,53,31,19)(11,40,23,35,44,60,43,61)(12,45,46,36,47,49,13,37)(25,52,33,58,57,50,26,55)(41,62,42,63,54,59,48,64);(1,4)(2,11,16,44)(3,18,22,51)(5,25)(6,33)(7,27)(8,28)(9,29)(10,35)(12,41,47,54)(13,48,46,42)(14,43,15,23)(17,40)(19,50,20,52)(21,24,34,56)(26,31)(30,55,53,58)(32,57)(36,59)(37,62)(38,60)(39,61)(45,63)(49,64);(1,5,8,31,9,32,7,6)(2,12,15,46,16,47,14,13)(3,19,21,30,22,20,34,53)(4,25,28,26,29,57,27,33)(10,36,17,45,39,37,38,49)(11,41,23,42,44,54,43,48)(18,50,24,55,51,52,56,58)(35,59,40,63,61,62,60,64);(1,6,7,32,9,31,8,5)(2,13,14,47,16,46,15,12)(3,20,21,53,22,19,34,30)(4,26,27,25,29,33,28,57)(10,37,17,49,39,36,38,45)(11,42,43,41,44,48,23,54)(18,50,24,55,51,52,56,58)(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,e^c=ei

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, 6375316);