#489402

order	512
nilpotency class	4
isoclinism	isoclinic class of #489398
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 ◁ #2965 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

778 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)(11,20)(12,24)(13,29)(14,19)(15,21)(16,34)(17,28)(18,27)(22,31)(23,35)(25,38)(26,36)(30,37)(32,39)(33,40);(3,7)(5,8)(9,11)(10,15)(12,25)(13,30)(14,22)(16,35)(17,37)(18,38)(19,32)(20,24)(21,28)(23,27)(26,40)(29,33)(31,36)(34,39);(1,2)(3,8)(4,6)(5,7)(12,26)(16,17)(20,31)(21,39)(24,36)(25,40)(28,34)(35,37);(9,12,27,34,14,26,13,28)(10,16,29,24,19,17,18,36)(11,21,30,40,22,39,23,25)(15,31,38,37,32,20,33,35)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 396598);