#144914

order	512
nilpotency class	4
isoclinism	isoclinic class of #142406
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1616
commutator, Frattini, cosocle	C₂²×M₁₆	C₂³
prime radical	C₂×#469	C₂²
LCS	G ▷ C₂²×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×M₁₆ ◁ G
LECS	G ▷ C₂²×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 407 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) = 64

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 52110);