#145906

order	512
nilpotency class	4
isoclinism	isoclinic class of #143954
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1687
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂	#10587
prime radical	#3147	C₂²
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

242 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 242 conjugacy classes of subgroups — too many to draw. Only its 74 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:

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

Polycyclic presentation:

2^9|a^2=de,b^2=d,d^2=g,e^2=h,f^2=gi,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=egi,f^a=fg,g^a=gi,h^a=hi,c^b=cf,e^b=ei,d^c=dg,e^c=eh,f^c=fg,g^c=gi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 53102);