#491116

order	512
nilpotency class	4
isoclinism	isoclinic class of #491113
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#466
commutator, Frattini, cosocle	C₄×C₈	C₂⁴
prime radical	#19332	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2931 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

447 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)(4,6)(7,8)(10,16)(11,22)(12,14)(13,26)(17,27)(18,33)(19,30)(20,29)(21,31)(23,25)(24,38)(28,34)(35,40)(36,39);(9,10,17,29,12,20,13,16,15,21,26,30,14,19,27,31)(11,18,32,40,22,34,24,28,25,35,37,33,23,36,38,39);(2,5)(3,8)(6,7)(9,11)(10,18)(12,22)(13,24)(14,23)(15,25)(16,28)(17,32)(19,36)(20,34)(21,35)(26,37)(27,38)(29,40)(30,33)(31,39);(2,6)(5,7)(9,12,15,14)(10,19,21,20)(11,23,25,22)(13,26,27,17)(16,29,31,30)(18,34,35,36)(24,32,38,37)(28,33,39,40)

Polycyclic presentation:

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

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, 398312);