#489192

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
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

788 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)(11,20)(12,17)(13,25)(14,29)(15,19)(16,22)(18,27)(21,37)(23,38)(24,36)(26,31)(28,35)(30,34)(32,40)(33,39);(3,7)(5,8)(9,11)(10,16)(12,21)(13,26)(14,30)(15,23)(17,31)(18,34)(19,32)(20,25)(22,27)(24,40)(28,39)(29,33)(35,38)(36,37);(1,2,4,6)(3,8,7,5)(9,12,15,14)(10,17,19,29)(11,21,23,30)(13,27,28,24)(16,31,32,33)(18,35,36,25)(20,34,38,37)(22,39,40,26);(9,12,15,14)(10,17,19,29)(11,21,23,30)(13,24,28,27)(16,31,32,33)(18,25,36,35)(20,37,38,34)(22,26,40,39)

Polycyclic presentation:

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

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, 396388);