#153246

order	512
nilpotency class	5
isoclinism	isoclinic class of #153246
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#13380
commutator, Frattini, cosocle	C₂²×C₁₆	C₂³
prime radical	C₂×#2221	C₂
LCS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂×#131 ◁ C₂×#504 ◁ G
LECS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 266 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:

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

Polycyclic presentation:

2^9|c^2=d,d^2=i,e^2=gh,g^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=eg,g^a=gh,h^a=hi,c^b=cf,d^b=di,e^c=egi,f^c=fi,g^c=gh,h^c=hi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72320657, #72321417, #72321446
maximal normal subgroup of	#72320657, #72321417, #72321446

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 60442);