#489975

order	512
nilpotency class	4
isoclinism	isoclinic class of #489719
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 ◁ #2966 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

744 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:

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

Polycyclic presentation:

2^9|d^2=hi,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,d^b=di,e^b=eg,f^b=fh,g^b=gi,d^c=di,e^c=eh,f^c=fh

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 397171);