#145926

order	512
nilpotency class	4
isoclinism	isoclinic class of #143951
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1667
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

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

normal subgroups only

This group has 275 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:

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

Polycyclic presentation:

2^9|a^2=de,b^2=dg,d^2=gi,e^2=h,f^2=g,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=egi,f^a=fgi,g^a=gi,h^a=hi,c^b=cf,d^c=dg,e^c=eh,f^c=fgi,g^c=gi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 53122);