#497806

order	512
nilpotency class	3
isoclinism	isoclinic class of #24986
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₈	#555
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	#22633
prime radical	C₂×#41	C₂²×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

367 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 405002);