#426460

order	512
nilpotency class	3
isoclinism	isoclinic class of #418473
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2673
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15353
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

758 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:

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

Polycyclic presentation:

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

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, 333656);