#489714

order	512
nilpotency class	4
isoclinism	isoclinic class of #489711
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

904 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:

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

Polycyclic presentation:

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

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, 396910);