#153243

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	#33604	C₂
LCS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂³×C₈ ◁ C₂²×M₃₂ ◁ G
LECS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 281 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:

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

Polycyclic presentation:

2^9|c^2=d,e^2=gh,g^2=hi,h^2=i,b^a=bd,c^a=ce,e^a=eg,f^a=fi,g^a=gh,h^a=hi,c^b=cf,e^b=ei,e^c=eg,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, 60439);