#140638

order	512
nilpotency class	4
isoclinism	isoclinic class of #13229
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1964
commutator	C₂²×C₄	C₂²×C₈
Frattini, cosocle	C₂²×M₁₆	C₂³
socle	C₂	#9650
prime radical	C₂⁴	#131
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

351 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 351 conjugacy classes of subgroups — too many to draw. Only its 102 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfg,c^2=e,d^2=i,e^2=i,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,g^a=gi,d^b=dg,f^b=fgi,g^b=gi,f^d=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, 47834);