#149254

order	512
nilpotency class	4
isoclinism	isoclinic class of #13313
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

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	C₂⁴	#135
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

536 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

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

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, 56450);