#497122

order	512
nilpotency class	3
isoclinism	isoclinic class of #22284
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2328
commutator	C₂×C₄	C₂³×C₈
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	#22260
prime radical	#3147	C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

576 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 404318);