#479394

order	512
nilpotency class	3
isoclinism	isoclinic class of #479394
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#17204
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

785 conjugacy classes of subgroups; only the 160 normal subgroups are shown.

normal subgroups only

This group has 785 conjugacy classes of subgroups — too many to draw. Only its 160 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 386590);