#462294

order	512
nilpotency class	3
isoclinism	isoclinic class of #458124
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2731
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#16425
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

666 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:

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

Polycyclic presentation:

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

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, 369490);