#489717

order	512
nilpotency class	4
isoclinism	isoclinic class of #489711
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	#61084	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2931 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

935 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)(9,10,15,20)(11,17,23,37)(12,18,14,30)(13,26,29,40)(16,31,34,22)(19,25,38,28)(21,35,24,36)(27,32,39,33);(2,6)(3,4)(7,8)(9,11)(10,16)(12,21)(13,27)(14,24)(15,23)(17,26)(18,32)(19,36)(20,34)(22,28)(25,31)(29,39)(30,33)(35,38)(37,40);(2,7)(3,4)(6,8)(9,11)(10,17)(12,24)(13,22)(14,21)(15,23)(16,26)(18,36)(19,32)(20,37)(25,39)(27,28)(29,31)(30,35)(33,38)(34,40);(2,8)(6,7)(9,12,15,14)(10,18,20,30)(11,21,23,24)(13,25,29,28)(16,32,34,33)(17,35,37,36)(19,26,38,40)(22,27,31,39)

Polycyclic presentation:

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

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 396913);