#510028

order	512
nilpotency class	4
isoclinism	isoclinic class of #509155
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	D₄×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#33589	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1772 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

548 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:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=ei,c^2=e,f^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=def,c^b=cg,d^b=de,g^b=gi,f^c=fh,g^c=gi,h^c=hi,f^d=fh,h^d=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, 417224);