#135850

order	512
nilpotency class	4
isoclinism	isoclinic class of #135850
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1854
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂	#8834
prime radical	C₂×#41	C₄⋊C₄
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₈ ◁ C₄×#131 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

251 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 251 conjugacy classes of subgroups — too many to draw. Only its 90 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 — 3 generators on 64 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=f,c^2=g,d^2=gi,f^2=h,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,g^a=gi,c^b=ch,d^b=dg,g^b=gi,e^c=ei,f^c=fi

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, 43046);