#499382

order	512
nilpotency class	4
isoclinism	isoclinic class of #498807
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	C₂²×C₈	C₂⁴
prime radical	C₂²×#41	D₄
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

682 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=f,c^2=e,f^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=di,c^b=cg,g^b=gi,f^c=fh,g^c=gi,h^c=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, 406578);