#489201

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 ◁ #2960 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

775 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:

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

Polycyclic presentation:

2^9|c^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, 396397);