#147708

order	512
nilpotency class	4
isoclinism	isoclinic class of #13313
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#466
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
prime radical	C₂²×#472	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1080 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:

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

Polycyclic presentation:

2^9|a^2=g,d^2=h,e^2=i,f^2=i,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,h^a=hi,c^b=cf,d^b=dhi,e^b=ei,f^b=fi,g^b=gi,h^b=hi,d^c=di,e^c=ei,f^c=fi,g^c=gi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 54904);