#436736

order	512
nilpotency class	3
isoclinism	isoclinic class of #420416
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2555
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15569
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 897 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 343932);