#153344

order	512
nilpotency class	5
isoclinism	isoclinic class of #153105
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2206
commutator, Frattini, cosocle	C₂²×C₁₆	C₂³
prime radical	C₂²×D₁₆	C₂²
LCS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆ ◁ G
LECS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

226 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 226 conjugacy classes of subgroups — too many to draw. Only its 68 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:

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

Polycyclic presentation:

2^9|a^2=de,b^2=dg,d^2=gh,e^2=g,f^2=gi,g^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=egh,f^a=fgh,g^a=gh,h^a=hi,c^b=cf,f^b=fi,d^c=dg,e^c=egh,f^c=fghi,g^c=gh,h^c=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 60540);