#509422

order	512
nilpotency class	4
isoclinism	isoclinic class of #509097
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	#33612	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1764 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

500 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:

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

Polycyclic presentation:

2^9|a^2=e,c^2=ei,f^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=def,f^a=fh,h^a=hi,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

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 416618);