#506228

order	512
nilpotency class	4
isoclinism	isoclinic class of #500681
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#459
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#33519	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

709 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)(4,6)(5,8)(9,10)(11,17)(12,26)(13,19)(14,29)(15,21)(16,22)(18,27)(20,28)(23,34)(24,36)(25,37)(30,39)(31,38)(32,33)(35,40);(1,3,4,8)(2,5,6,7)(9,11,15,23,16,12,13,25)(10,17,21,34,22,26,19,37)(14,24,31,30,32,27,28,40)(18,20,35,29,36,38,39,33);(9,12)(10,18)(11,16)(13,25)(14,30)(15,23)(17,33)(19,35)(20,37)(21,39)(22,36)(24,31)(26,29)(27,28)(32,40)(34,38);(2,6)(5,7)(9,13,16,15)(10,19,22,21)(11,23,12,25)(14,28,32,31)(17,34,26,37)(18,35,36,39)(20,33,38,29)(24,30,27,40)

Polycyclic presentation:

2^9|b^2=eg,d^2=i,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,d^b=di,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, 413424);