#510054

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	C₂²×D₈	C₂³
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1772 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

481 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)(9,10,18,33,16,20,31,35,14,22,15,23,17,24,12,29)(11,25,26,36,32,37,34,21,13,19,27,40,28,38,30,39);(1,3,5,8)(2,6,7,4)(9,11)(10,19)(12,30)(13,14)(15,27)(16,32)(17,28)(18,26)(20,38)(21,29)(22,25)(23,36)(24,37)(31,34)(33,40)(35,39);(9,12)(10,20)(11,26)(13,27)(14,31)(15,16)(17,18)(19,25)(21,36)(22,24)(28,34)(29,35)(30,32)(39,40);(1,4,5,6)(2,3,7,8)(9,13,14,11)(10,21,22,39)(12,27,31,26)(15,30,18,34)(16,32,17,28)(19,35,25,29)(20,36,24,40)(23,38,33,37)

Polycyclic presentation:

2^9|a^2=ef,b^2=ei,d^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, 417250);