#489385

order	512
nilpotency class	4
isoclinism	isoclinic class of #489385
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

761 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)(11,21)(12,26)(13,17)(14,31)(15,20)(16,23)(18,28)(19,29)(22,38)(24,39)(25,33)(27,34)(30,36)(32,37)(35,40);(3,7)(5,8)(9,11)(10,16)(12,27)(13,22)(14,32)(15,24)(17,21)(18,37)(19,38)(20,34)(23,28)(25,29)(26,33)(30,40)(31,35)(36,39);(1,2)(3,8)(4,6)(5,7)(9,10)(11,16)(12,17)(13,26)(14,19)(15,20)(18,30)(21,27)(22,33)(23,39)(24,34)(25,35)(28,36)(29,31)(32,38)(37,40);(9,12,14,13,15,28,29,30)(10,17,19,26,20,36,31,18)(11,22,25,27,24,40,32,23)(16,33,35,21,34,37,38,39)

Polycyclic presentation:

2^9|d^2=g,e^2=gi,f^2=hi,g^2=i,b^a=be,c^a=cf,d^a=dg,e^a=eg,f^a=fhi,g^a=gi,d^b=dg,e^b=eg,f^b=fh,g^b=gi,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, 396581);