#504885

order	512
nilpotency class	4
isoclinism	isoclinic class of #500857
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

573 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)(3,7)(4,6)(5,8)(9,10)(11,22)(12,27)(13,31)(14,19)(15,21)(16,25)(17,29)(18,33)(20,32)(23,39)(24,37)(26,30)(28,40)(34,35)(36,38);(1,3,4,8)(2,5,6,7)(9,11,23,38,14,27,13,25,15,28,33,37,32,26,34,17)(10,16,35,12,20,36,18,22,21,29,31,30,19,24,39,40);(2,6)(5,7)(9,10)(11,24)(12,25)(13,18)(14,20)(15,21)(16,27)(17,30)(19,32)(22,37)(23,35)(26,29)(28,36)(31,33)(34,39)(38,40);(2,6)(5,7)(9,12,23,36,14,22,13,29,15,30,33,24,32,40,34,16)(10,17,35,11,20,38,18,27,21,25,31,28,19,37,39,26)

Polycyclic presentation:

2^9|b^2=ehi,d^2=efhi,e^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,h^a=hi,c^b=cg,g^b=gi,d^c=di,g^c=gi

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, 412081);