#468974

order	512
nilpotency class	3
isoclinism	isoclinic class of #455576
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2730
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#16637
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

667 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:

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

Polycyclic presentation:

2^9|a^2=eg,b^2=fg,c^2=g,d^2=f,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,h^b=hi,d^c=de,e^c=ei,h^c=hi,e^d=ei

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 376170);