#502752

order	512
nilpotency class	4
isoclinism	isoclinic class of #501256
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

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

836 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)(9,10,14,18)(11,15,20,30)(12,23,27,16)(13,17,25,34)(19,37,38,29)(21,40,39,31)(22,32,28,36)(24,35,26,33);(2,5)(6,7)(9,11)(10,15)(12,19)(13,28)(14,20)(16,29)(17,36)(18,30)(21,26)(22,25)(23,37)(24,39)(27,38)(31,35)(32,34)(33,40);(11,19,20,38)(12,24)(13,25)(15,29,30,37)(16,33)(17,34)(21,28,39,22)(23,35)(26,27)(31,36,40,32);(1,3,4,8)(2,6,5,7)

Polycyclic presentation:

2^9|a^2=i,c^2=f,d^2=e,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=ghi,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, 409948);