#508236

order	512
nilpotency class	4
isoclinism	isoclinic class of #508016
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	#3061	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₄×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

722 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:

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

Polycyclic presentation:

2^9|b^2=i,c^2=ei,d^2=e,f^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,f^a=fh,h^a=hi,c^b=cg,g^b=gi,d^c=de,f^c=fh,g^c=gi,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, 415432);