#153256

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

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2233
commutator, Frattini, cosocle	C₂²×C₁₆	C₂³
prime radical	C₂⁴	SD₃₂
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

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

normal subgroups only

This group has 268 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) = 68
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 68 points:

(5,6,11,16)(7,17,18,46)(8,22,23,56)(9,28,31,64)(10,32,30,15)(12,34,35,19)(13,38,39,26)(14,42,44,25)(20,54,50,68)(21,55,33,49)(24,60,52,43)(27,47,59,58)(29,48,65,40)(36,61,67,63)(37,53,45,66)(41,62,57,51);(1,2)(3,4)(8,23)(13,39)(17,46)(19,34)(22,56)(24,52)(26,38)(27,59)(40,48)(41,57)(47,58)(49,55)(51,62)(53,66)(54,68)(61,63);(2,4)(5,7,10,33,11,18,30,21)(6,12,15,45,16,35,32,37)(8,24,27,63,23,52,59,61)(9,29,25,20,31,65,42,50)(13,40,41,54,39,48,57,68)(14,43,28,36,44,60,64,67)(17,47,55,22,46,58,49,56)(19,51,66,26,34,62,53,38)

Polycyclic presentation:

2^9|a^2=i,c^2=dh,e^2=gh,g^2=hi,h^2=i,b^a=bd,c^a=ce,e^a=eg,g^a=gh,h^a=hi,c^b=cf,e^b=ei,d^c=di,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, 60452);