#489021

order	512
nilpotency class	4
isoclinism	isoclinic class of #488920
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
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2931 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

715 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:

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

Polycyclic presentation:

2^9|b^2=h,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,d^c=di,e^c=eh,f^c=fhi

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, 396217);