#153259

order	512
nilpotency class	5
isoclinism	isoclinic class of #153259
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#13381
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

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

normal subgroups only

This group has 267 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)(3,15)(4,6)(5,23)(7,25)(10,33)(11,37)(12,14)(13,43)(16,49)(17,47)(18,51)(19,46)(20,48)(21,53)(22,29)(24,28)(26,54)(27,55)(30,34)(31,40)(35,38)(36,45)(39,63)(41,62)(44,61)(50,58)(52,57)(56,60)(59,64);(1,2)(3,16)(4,12)(5,11)(6,9)(7,13)(8,14)(10,17)(15,45)(18,44)(19,36)(20,30)(21,39)(22,42)(23,40)(24,56)(25,32)(26,31)(27,41)(28,59)(29,43)(33,48)(34,46)(35,50)(37,53)(38,57)(47,49)(51,64)(52,60)(54,62)(55,63)(58,61);(1,3)(2,10,14,36)(4,19)(5,24)(6,17)(7,28)(8,20)(9,30,12,16)(11,38,41,64)(13,44,43,60)(15,23)(18,25)(21,51)(22,52)(26,57)(27,58)(29,50)(31,61,39,56)(32,35,42,59)(33,62,45,37)(34,63,49,40)(46,53)(47,54)(48,55)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72320815, #72321372, #72321425
maximal normal subgroup of	#72320815, #72321372, #72321425

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 60455);