#413505

order	512
nilpotency class	3
isoclinism	isoclinic class of #401030
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2541
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15148
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1005 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=ef,c^2=ef,d^2=h,e^2=i,g^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^b=gi,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, 320701);