#492251

order	512
nilpotency class	4
isoclinism	isoclinic class of #492236
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	C₄×C₈	C₂⁴
prime radical	#19953	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈×D₈ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

438 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=i,e^2=g,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=di,e^a=eg,f^a=fh,h^a=hi,c^b=cg,d^b=de,e^b=eg,f^c=fh,h^c=hi,e^d=eg

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 399447);