#413020

order	512
nilpotency class	3
isoclinism	isoclinic class of #400512
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2491
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15139
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

663 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=ef,c^2=e,d^2=h,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,h^a=hi,c^b=cg,e^b=ei,f^b=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, 320216);