#153386

order	512
nilpotency class	5
isoclinism	isoclinic class of #153386
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#13507
commutator, Frattini, cosocle	C₂²×C₁₆	C₂³
prime radical	C₂³	#508
LCS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #405 ◁ #2108 ◁ G
LECS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

156 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 156 conjugacy classes of subgroups — too many to draw. Only its 66 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:

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

Polycyclic presentation:

2^9|a^2=eh,b^2=d,c^2=d,d^2=i,e^2=i,f^2=gh,g^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=di,f^a=fi,c^b=cf,e^b=ei,f^b=fg,g^b=gh,h^b=hi,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	#72320657, #72321384, #72321450
maximal normal subgroup of	#72320657, #72321384, #72321450

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 60582);