#462117

order	512
nilpotency class	3
isoclinism	isoclinic class of #458124
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2731
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#16418
prime radical	C₂⁴	2₋¹⁺⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

557 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=f,c^2=f,d^2=e,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,d^c=de,e^c=ei,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, 369313);