#153267

order	512
nilpotency class	5
isoclinism	isoclinic class of #153090
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2184
commutator, Frattini, cosocle	C₂²×C₁₆	C₂³
prime radical	#33716	C₂
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

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

normal subgroups only

This group has 313 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:

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

Polycyclic presentation:

2^9|c^2=d,e^2=gh,f^2=gh,g^2=hi,h^2=i,b^a=bd,c^a=ce,e^a=eg,f^a=fgi,g^a=gh,h^a=hi,c^b=cf,e^b=egi,f^b=fg,g^b=gh,h^b=hi,e^c=eg,f^c=fg,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, 60463);