#153261

order	512
nilpotency class	5
isoclinism	isoclinic class of #153261
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

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

normal subgroups only

This group has 268 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)(5,23)(6,26)(10,34)(11,38)(12,41)(13,20)(15,46)(16,45)(17,44)(18,51)(19,43)(21,54)(22,30)(24,58)(25,29)(27,55)(28,57)(31,36)(32,40)(35,47)(37,39)(42,63)(48,59)(49,61)(50,60)(52,56)(62,64);(1,2)(3,15)(4,20)(5,24)(6,12)(7,9)(8,13)(10,17)(11,28)(14,36)(16,35)(18,48)(19,31)(21,32)(22,53)(23,56)(25,39)(26,33)(27,52)(29,62)(30,41)(34,43)(37,50)(38,55)(40,57)(42,49)(44,47)(45,46)(51,64)(54,58)(59,61)(60,63);(1,3)(2,10,13,15)(4,19)(5,25)(6,29)(7,16)(8,17)(9,31,20,35)(11,39,24,59)(12,42,41,62)(14,23)(18,26)(21,51)(22,50)(27,60)(28,61)(30,49)(32,63,52,64)(33,37,53,48)(34,58,46,38)(36,56,47,40)(43,54)(44,57)(45,55)

Polycyclic presentation:

2^9|c^2=dh,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^b=ei,d^c=di,e^c=eg,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, #72321372, #72321435
maximal normal subgroup of	#72320657, #72321372, #72321435

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 60457);