#404688

order	512
nilpotency class	3
isoclinism	isoclinic class of #404687
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#14899
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 700 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 — 4 generators on 64 points:

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

Polycyclic presentation:

2^9|a^2=efi,b^2=ef,c^2=e,d^2=g,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,e^b=ei,f^b=fi,f^c=fi,h^d=hi

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 311884);