#152794

order	512
nilpotency class	3
isoclinism	isoclinic class of #11247
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₈	#387
commutator	C₂²×C₄	C₂²×C₈
Frattini, cosocle	C₂²×C₁₆	C₂³
socle	C₂	#13105
prime radical	C₂³	#504
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂²×C₁₆ ◁ G
LECS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

166 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 166 conjugacy classes of subgroups — too many to draw. Only its 84 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=f,c^2=d,d^2=i,f^2=g,g^2=h,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=ei,c^b=ch,d^b=di,e^c=ei,f^c=fi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 59990);