#486311

order	512
nilpotency class	4
isoclinism	isoclinic class of #486306
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#447
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#62258	C₂
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#435 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

900 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:

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

Polycyclic presentation:

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

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, 393507);