#153395

order	512
nilpotency class	5
isoclinism	isoclinic class of #153213
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2226
commutator, Frattini, cosocle	C₂²×C₁₆	C₂³
prime radical	C₂³	#509
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

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

normal subgroups only

This group has 172 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:

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

Polycyclic presentation:

2^9|a^2=eh,b^2=d,c^2=d,d^2=i,e^2=ghi,g^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=di,c^b=cf,e^c=eg,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, 60591);