#510056

order	512
nilpotency class	4
isoclinism	isoclinic class of #509163
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	C₂²×D₈	C₂³
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1772 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

479 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:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=e,d^2=e,f^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=defi,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

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 417252);