#153385

order	512
nilpotency class	5
isoclinism	isoclinic class of #153329
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2254
commutator, Frattini, cosocle	C₂²×C₁₆	C₂³
socle	C₂	#13506
prime radical	C₂³	#510
LCS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×#346 ◁ G
LECS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

165 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 165 conjugacy classes of subgroups — too many to draw. Only its 70 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) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 64 points:

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

Polycyclic presentation:

2^9|a^2=e,b^2=d,c^2=dh,e^2=i,f^2=ghi,g^2=hi,h^2=i,b^a=bd,c^a=ce,f^a=fi,c^b=cf,f^b=fg,g^b=gh,h^b=hi,d^c=di,e^c=ei,f^c=fg,g^c=gh,h^c=hi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72321265, #72321372, #72321384
maximal normal subgroup of	#72321265, #72321372, #72321384

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 60581);