#479879

order	512
nilpotency class	3
isoclinism	isoclinic class of #479233
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2785
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#17300
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

499 conjugacy classes of subgroups; only the 144 normal subgroups are shown.

normal subgroups only

This group has 499 conjugacy classes of subgroups — too many to draw. Only its 144 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 387075);