#503059

order	512
nilpotency class	4
isoclinism	isoclinic class of #501060
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	#60682	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

752 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 410255);