#490480

order	512
nilpotency class	4
isoclinism	isoclinic class of #490479
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#466
commutator, Frattini, cosocle	C₄×C₈	C₂⁴
prime radical	#19190	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2931 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

518 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=i,b^2=eg,e^2=gi,f^2=h,g^2=i,b^a=be,c^a=cf,e^a=eg,f^a=fh,g^a=gi,d^b=di,d^c=di,f^c=fh

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, 397676);