#489057

order	512
nilpotency class	4
isoclinism	isoclinic class of #12324
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#447
commutator, Frattini, cosocle	C₄×C₈	C₂⁴
socle	C₂²	C₂×#447
prime radical	#33579	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×#446 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

713 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)(11,22)(12,18)(13,29)(14,20)(15,21)(16,24)(17,25)(19,32)(23,40)(26,37)(27,35)(28,39)(30,34)(31,36)(33,38);(1,2,4,6)(3,8,7,5)(9,11)(10,16)(12,23)(13,30)(14,26)(15,27)(17,21)(18,34)(19,37)(20,36)(22,29)(24,32)(25,28)(31,33)(35,38)(39,40);(9,11)(10,17)(12,23)(13,31)(14,26)(15,27)(16,21)(18,36)(19,37)(20,34)(22,29)(24,28)(25,32)(30,33)(35,38)(39,40);(9,12,15,14)(10,18,21,20)(11,23,27,26)(13,28,33,32)(16,34,17,36)(19,29,39,38)(22,40,35,37)(24,30,25,31)

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,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, 396253);