#153406

order	512
nilpotency class	5
isoclinism	isoclinic class of #153255
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2233
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×#346 ◁ G
LECS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

229 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 229 conjugacy classes of subgroups — too many to draw. Only its 72 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:

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

Polycyclic presentation:

2^9|a^2=dh,b^2=deg,d^2=ghi,e^2=ghi,g^2=hi,h^2=i,b^a=bd,c^a=ce,e^a=ei,f^a=fi,c^b=cf,d^b=dg,e^b=eg,g^b=gh,h^b=hi,d^c=dgi,e^c=egi,g^c=gh,h^c=hi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 60602);