#489196

order	512
nilpotency class	4
isoclinism	isoclinic class of #488912
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#447
commutator, Frattini, cosocle	C₄×C₈	C₂⁴
prime radical	#61401	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#446 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

783 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)(3,7)(6,8)(9,10,15,20)(11,21,24,38)(12,17,14,18)(13,26,29,35)(16,31,34,23)(19,25,37,28)(22,39,30,36)(27,32,40,33);(3,7)(5,8)(9,11)(10,16)(12,22)(13,27)(14,30)(15,24)(17,32)(18,33)(19,36)(20,34)(21,26)(23,28)(25,31)(29,40)(35,38)(37,39);(1,2,4,6)(3,8,7,5)(9,12,15,14)(10,17,20,18)(11,22,24,30)(13,28,29,25)(16,32,34,33)(19,35,37,26)(21,36,38,39)(23,40,31,27);(9,12,15,14)(10,18,20,17)(11,22,24,30)(13,25,29,28)(16,33,34,32)(19,35,37,26)(21,36,38,39)(23,27,31,40)

Polycyclic presentation:

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

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, 396392);