#144759

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	C₂⁴	C₂×SD₁₆
LCS	G ▷ C₂²×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×M₁₆ ◁ G
LECS	G ▷ C₂²×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

350 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 350 conjugacy classes of subgroups — too many to draw. Only its 66 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=d,d^2=i,e^2=gi,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=di,e^a=eg,f^a=fhi,g^a=gi,c^b=cf,e^b=eh,e^c=egi,f^c=fi,g^c=gi,h^c=hi,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, 51955);