#146841

order	512
nilpotency class	3
isoclinism	isoclinic class of #465
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₈ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₈	C₂×D₄
commutator	C₂×C₄	C₂×C₄×C₈
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂³	C₈×D₄
prime radical	C₂⁴	#131
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₈ ◁ C₂³×C₁₆ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

542 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)(5,6,7,8)(9,10,14,19,16,20,12,29,17,21,15,32,13,31,30,40)(11,22,25,36,27,35,23,38,28,37,26,39,24,18,33,34);(9,11,15,26,16,27,30,33,17,28,14,25,13,24,12,23)(10,18,19,34,20,22,29,36,21,35,32,38,31,37,40,39);(2,4)(9,12,13,14,17,30,16,15)(10,19,20,29,21,32,31,40)(11,23,24,25,28,33,27,26)(18,34,22,36,35,38,37,39)

Polycyclic presentation:

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

Conjugacy classes & character table

176 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 54037);