#486331

order	512
nilpotency class	4
isoclinism	isoclinic class of #486327
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
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #2929 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

933 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

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

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, 393527);