#489066

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
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

698 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)(11,23)(12,25)(13,18)(14,33)(15,36)(16,21)(17,22)(19,32)(20,31)(24,38)(26,35)(27,39)(28,37)(29,34)(30,40);(1,2,4,6)(3,8,7,5)(9,11)(10,12)(13,26)(14,34)(15,37)(16,24)(17,27)(18,29)(19,38)(20,35)(21,28)(22,30)(23,33)(25,32)(31,40)(36,39);(9,12)(10,11)(13,28)(14,35)(15,38)(16,29)(17,30)(18,24)(19,37)(20,34)(21,26)(22,27)(23,31)(25,36)(32,39)(33,40);(9,13,17,16)(10,18,22,21)(11,24,27,26)(12,28,30,29)(14,31,15,32)(19,33,20,36)(23,38,39,35)(25,37,40,34)

Polycyclic presentation:

2^9|b^2=gh,d^2=i,e^2=gi,f^2=h,g^2=i,h^2=i,b^a=be,c^a=cf,e^a=eg,f^a=fhi,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=fhi,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, 396262);