#488911

order	512
nilpotency class	4
isoclinism	isoclinic class of #12324
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

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

1001 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:

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

Polycyclic presentation:

2^9|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, 396107);