#145952

order	512
nilpotency class	4
isoclinism	isoclinic class of #144059
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1677
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂×#493	C₂²
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

269 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 269 conjugacy classes of subgroups — too many to draw. Only its 76 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

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:

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

Polycyclic presentation:

2^9|a^2=deh,b^2=d,d^2=g,e^2=h,f^2=g,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eg,f^a=fgi,g^a=gi,h^a=hi,c^b=cf,e^b=ei,d^c=dgi,e^c=eh,f^c=fgi,g^c=gi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 53148);