#146877

order	512
nilpotency class	4
isoclinism	isoclinic class of #2184
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂×D₈
commutator	C₂×C₈	C₂×C₄²
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂³	C₄×D₈
prime radical	C₂⁴	#135
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×C₈ ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

526 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)(6,7)(9,10)(11,19)(12,25)(13,18)(14,30)(15,20)(16,23)(17,22)(21,29)(24,33)(26,40)(27,36)(28,32)(31,37)(34,38)(35,39);(5,6,8,7)(9,11,15,24,13,23,29,14)(10,16,20,30,18,19,21,33)(12,22,27,39,26,38,28,31)(17,32,35,25,34,36,37,40);(5,7,8,6)(9,12,13,26)(10,17,18,34)(11,22,23,38)(14,31,24,39)(15,27,29,28)(16,32,19,36)(20,35,21,37)(25,33,40,30)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=d,e^2=hi,g^2=i,h^2=i,b^a=bd,c^a=ce,e^a=eh,h^a=hi,e^c=eh,h^c=hi

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 54073);