#489375

order	512
nilpotency class	4
isoclinism	isoclinic class of #489373
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

840 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)(11,16)(12,15)(13,27)(14,19)(17,24)(18,25)(20,35)(21,28)(22,39)(23,34)(26,30)(29,40)(31,36)(32,38)(33,37);(3,7)(5,8)(10,15)(12,24)(13,25)(16,28)(17,18)(19,30)(20,36)(21,38)(22,37)(26,40)(27,29)(31,39)(32,33)(34,35);(1,2)(3,8)(4,6)(5,7)(9,11)(10,16)(12,21)(13,22)(14,23)(15,28)(17,32)(18,33)(19,34)(20,26)(24,38)(25,37)(27,39)(29,31)(30,35)(36,40);(9,12,25,40,14,26,13,24)(10,17,27,30,19,29,18,15)(11,20,37,38,23,21,22,36)(16,31,39,28,34,32,33,35)

Polycyclic presentation:

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

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, 396571);