#138144

order	512
nilpotency class	4
isoclinism	isoclinic class of #138132
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1874
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂×#472	C₄
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₄×#131 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

428 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfgh,d^2=ghi,e^2=i,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dg,d^b=dh,e^b=ei,f^b=fh,h^b=hi,d^c=di,e^c=ei,f^c=fi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 45340);