#153364

order	512
nilpotency class	5
isoclinism	isoclinic class of #153364
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#13482
commutator, Frattini, cosocle	C₂²×C₁₆	C₂³
prime radical	C₂²×D₁₆	C₂²
LCS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #405 ◁ #2108 ◁ G
LECS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

198 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 198 conjugacy classes of subgroups — too many to draw. Only its 62 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:

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

Polycyclic presentation:

2^9|a^2=eh,c^2=d,f^2=gh,g^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=di,c^b=cf,f^b=fg,g^b=gh,h^b=hi,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	#72320657, #72321401, #72321425
maximal normal subgroup of	#72320657, #72321401, #72321425

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 60560);