#489045

order	512
nilpotency class	4
isoclinism	isoclinic class of #488901
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
prime radical	#33579	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2960 ◁ 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) = 256

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

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

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, 396241);