#501495

order	512
nilpotency class	4
isoclinism	isoclinic class of #501014
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#459
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#33582	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₈×C₄⋊C₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

629 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)(9,10,15,19,16,20,13,14)(11,17,31,35,23,33,22,29)(12,24,27,18,28,30,26,34)(21,36,39,32,38,25,37,40);(2,5)(6,7)(9,11)(10,17)(12,25)(13,22)(14,29)(15,31)(16,23)(18,21)(19,35)(20,33)(24,37)(26,32)(27,40)(28,36)(30,39)(34,38);(9,12,16,28)(10,18,20,34)(11,21,23,38)(13,27,15,26)(14,30,19,24)(17,32,33,40)(22,39,31,37)(25,35,36,29);(1,3,4,8)(2,6,5,7)

Polycyclic presentation:

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

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, 408691);