#145358

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

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1667
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂×#135	D₄
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

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

normal subgroups only

This group has 310 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) = 128

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

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

Polycyclic presentation:

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

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, 52554);