#506291

order	512
nilpotency class	4
isoclinism	isoclinic class of #500676
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#459
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#33580	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

674 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)(3,7)(4,5)(6,8)(9,10)(11,15)(12,16)(13,18)(14,20)(17,26)(19,21)(22,31)(23,33)(24,30)(25,32)(27,35)(28,37)(29,38)(34,36)(39,40);(2,5)(6,7)(9,11,21,22,14,25,13,24)(10,15,19,31,20,32,18,30)(12,23,36,37,27,39,26,38)(16,33,34,28,35,40,17,29);(10,16)(11,22)(12,26)(13,21)(15,28)(17,19)(18,34)(20,35)(24,25)(27,36)(29,32)(30,40)(31,33)(37,38);(1,3,4,8)(2,6,5,7)

Polycyclic presentation:

2^9|b^2=gi,d^2=e,f^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,h^a=hi,c^b=cg,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, 413487);