#452813

order	512
nilpotency class	3
isoclinism	isoclinic class of #427241
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2471
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#16065
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

587 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 360009);