#503091

order	512
nilpotency class	4
isoclinism	isoclinic class of #501256
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

740 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)(9,10,13,19)(11,22,23,17)(12,25,28,40)(14,32,15,34)(16,35,31,21)(18,24,39,29)(20,26,36,27)(30,38,33,37);(2,5)(6,7)(9,11)(10,17)(12,26)(13,23)(14,33)(15,30)(16,29)(18,21)(19,22)(20,25)(24,31)(27,28)(32,38)(34,37)(35,39)(36,40);(9,12,27,11,16,30,15,24,13,28,26,23,31,33,14,29)(10,18,32,37,21,17,36,40,19,39,34,38,35,22,20,25);(1,3,4,8)(2,6,5,7)(10,19)(17,22)(18,39)(20,36)(21,35)(25,40)(32,34)(37,38)

Polycyclic presentation:

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

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, 410287);