#145555

order	512
nilpotency class	4
isoclinism	isoclinic class of #143943
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	#459
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₈² ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

350 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 350 conjugacy classes of subgroups — too many to draw. Only its 98 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) = 512

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 52751);