#490001

order	512
nilpotency class	4
isoclinism	isoclinic class of #489745
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

717 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)(11,19)(12,23)(13,29)(14,16)(15,21)(17,27)(18,26)(20,36)(22,37)(24,34)(25,38)(28,39)(30,33)(31,35)(32,40);(1,2)(3,6)(4,8)(5,7)(9,11)(10,15)(12,24)(13,30)(14,20)(16,31)(17,33)(18,34)(19,23)(21,27)(22,26)(25,37)(28,40)(29,32)(35,38)(36,39);(1,2)(3,6)(4,8)(5,7)(9,10)(11,15)(12,25)(13,18)(14,16)(17,28)(19,35)(20,31)(21,36)(22,32)(23,38)(24,37)(26,29)(27,39)(30,34)(33,40);(1,3,5,4)(2,6,7,8)(10,16)(11,20)(17,25)(18,29)(21,35)(22,30)(23,39)(24,40)

Polycyclic presentation:

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

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, 397197);