#510031

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	#33589	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1772 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

546 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)(9,10,14,20,15,21,33,39,17,22,34,31,16,12,25,28)(11,18,19,30,35,40,27,29,24,32,38,13,23,26,37,36);(1,3,5,7)(2,6,8,4)(9,11,17,24)(10,18,22,32)(12,26,21,40)(13,20,30,31)(14,19,34,38)(15,35,16,23)(25,37,33,27)(28,36,39,29);(1,4,5,6)(2,7,8,3)(9,12)(10,16)(11,18)(13,27)(14,31)(15,22)(17,21)(19,36)(20,34)(23,40)(24,32)(25,28)(26,35)(29,38)(30,37)(33,39);(2,8)(3,7)(9,13)(10,19)(11,20)(12,27)(14,32)(15,29)(16,36)(17,30)(18,34)(21,37)(22,38)(23,39)(24,31)(25,26)(28,35)(33,40)

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=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, 417227);