#498669

order	512
nilpotency class	3
isoclinism	isoclinic class of #387433
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2372
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	#23273
prime radical	#532	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

414 conjugacy classes of subgroups; only the 162 normal subgroups are shown.

normal subgroups only

This group has 414 conjugacy classes of subgroups — too many to draw. Only its 162 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 — 4 generators on 64 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,c^2=e,d^2=f,e^2=i,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,c^b=ch,d^b=deh,e^b=ei,g^b=gi,g^c=gi,g^d=gi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 405865);