#153405

order	512
nilpotency class	5
isoclinism	isoclinic class of #153213
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2226
commutator, Frattini, cosocle	C₂²×C₁₆	C₂³
prime radical	C₂×#508	C₂²
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

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

normal subgroups only

This group has 225 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:

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

Polycyclic presentation:

2^9|a^2=dh,b^2=degi,d^2=ghi,e^2=gh,g^2=hi,h^2=i,b^a=bd,c^a=ce,e^a=ei,c^b=cf,d^b=dg,e^b=eg,g^b=gh,h^b=hi,d^c=dg,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, 60601);