#140076

order	512
nilpotency class	4
isoclinism	isoclinic class of #2238
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₈	#481
commutator	C₂×C₈	C₂²×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂	#9534
prime radical	C₂³	#443
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ C₂×#346 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

200 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 90 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=f,c^2=dg,e^2=gi,f^2=h,g^2=i,h^2=i,b^a=bd,c^a=ce,e^a=eg,g^a=gi,e^b=ei,d^c=di,e^c=egi,g^c=gi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 47272);