#492511

order	512
nilpotency class	4
isoclinism	isoclinic class of #492504
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	D₄×D₈
commutator, Frattini, cosocle	C₄×C₈	C₂⁴
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈×D₈ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

513 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 399707);