#413784

order	512
nilpotency class	3
isoclinism	isoclinic class of #403107
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2547
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15156
prime radical	C₂⁴	2₋¹⁺⁴
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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 320980);