#394446

order	512
nilpotency class	3
isoclinism	isoclinic class of #394098
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2403
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×2₋¹⁺⁴	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1971 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 40 points:

(1,2,4,6)(3,7,8,5)(9,10,14,20)(11,23,13,17)(12,26,28,37)(15,22,16,21)(18,31,36,29)(19,25,33,30)(24,38,27,35)(32,39,34,40);(1,3,4,8)(2,5,6,7)(9,11,14,13)(10,17,20,23)(12,24,28,27)(15,30,16,25)(18,34,36,32)(19,22,33,21)(26,35,37,38)(29,40,31,39);(9,12)(10,18)(11,24)(13,27)(14,28)(15,31)(16,29)(17,32)(19,35)(20,36)(21,37)(22,26)(23,34)(25,40)(30,39)(33,38);(9,13,14,11)(10,19,20,33)(12,27,28,24)(15,30,16,25)(17,22,23,21)(18,35,36,38)(26,34,37,32)(29,40,31,39)

Polycyclic presentation:

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

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, 301642);