#489012

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×#436 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

732 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:

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

Polycyclic presentation:

2^9|b^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, 396208);