#149102

order	512
nilpotency class	3
isoclinism	isoclinic class of #2069
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₈	#396
commutator	C₂²×C₄	C₂²×C₈
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂	#11298
prime radical	C₂⁴	#131
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

423 conjugacy classes of subgroups; only the 168 normal subgroups are shown.

normal subgroups only

This group has 423 conjugacy classes of subgroups — too many to draw. Only its 168 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:

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

Polycyclic presentation:

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

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, 56298);