#500456

order	512
nilpotency class	4
isoclinism	isoclinic class of #499754
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#447
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#61251	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#436 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

792 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)(13,25)(16,31)(19,34)(22,39)(24,28)(26,36)(27,40)(32,33);(2,5)(6,7)(9,10)(11,20)(12,17)(13,26)(14,29)(15,18)(16,22)(19,28)(21,37)(23,38)(24,34)(25,36)(27,32)(30,35)(31,39)(33,40);(9,11,15,23)(10,16,18,31)(12,21,14,30)(13,27,25,40)(17,32,29,33)(19,35,34,37)(20,36,38,26)(22,24,39,28);(1,3,4,8)(2,6,5,7)(9,12,15,14)(10,17,18,29)(11,21,23,30)(13,24,25,28)(16,32,31,33)(19,26,34,36)(20,37,38,35)(22,27,39,40)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 407652);