#153408

order	512
nilpotency class	5
isoclinism	isoclinic class of #153408
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#13528
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

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

normal subgroups only

This group has 206 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:

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

Polycyclic presentation:

2^9|a^2=d,b^2=degh,d^2=gh,e^2=gh,f^2=i,g^2=hi,h^2=i,b^a=bd,c^a=ce,f^a=fi,c^b=cf,d^b=dg,e^b=egi,g^b=gh,h^b=hi,d^c=dg,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, #72321384, #72321425
maximal normal subgroup of	#72320657, #72321384, #72321425

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 60604);