#145018

order	512
nilpotency class	4
isoclinism	isoclinic class of #143324
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1629
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

387 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 387 conjugacy classes of subgroups — too many to draw. Only its 78 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:

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

Polycyclic presentation:

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

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, 52214);