#146343

order	512
nilpotency class	4
isoclinism	isoclinic class of #360
automorphism	?
homology	Abelianization (H₁) = C₄³ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	#102
commutator	C₈	C₄³
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂²	C₄×#102
prime radical	C₂⁴	#135
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂³×C₈ ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

652 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=g,d^2=hi,f^2=h,g^2=i,h^2=i,b^a=bd,d^a=dh,f^a=fh,h^a=hi,d^b=di

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 53539);