#152280

order	512
nilpotency class	5
isoclinism	isoclinic class of #13691
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2206
commutator	C₂×C₁₆	C₂²×C₄
Frattini, cosocle	C₂²×C₁₆	C₂³
socle	C₂	#12811
prime radical	C₂×#508	C₂²
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆ ◁ G
LECS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

269 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 269 conjugacy classes of subgroups — too many to draw. Only its 96 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=d,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,g^a=gh,h^a=hi,e^c=eg,g^c=gh,h^c=hi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 59476);