#486327

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

906 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)(11,24)(12,29)(13,19)(14,22)(15,21)(16,23)(17,26)(18,27)(20,33)(25,38)(28,40)(30,36)(31,35)(32,39)(34,37);(1,2)(3,5)(4,6)(7,8)(9,11)(10,17)(12,30)(13,25)(14,27)(15,34)(16,28)(18,29)(19,24)(20,37)(21,35)(22,38)(23,36)(26,33)(31,32)(39,40);(1,3)(2,5)(4,7)(6,8)(9,11)(10,18)(12,31)(13,26)(14,34)(15,27)(16,28)(17,29)(19,38)(20,36)(21,40)(22,24)(23,37)(25,33)(30,32)(35,39);(9,12,15,13,16,33,14,32)(10,19,22,29,23,39,21,20)(11,25,27,30,28,31,34,26)(17,18,35,24,36,37,38,40)

Polycyclic presentation:

2^9|d^2=g,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, 393523);