#140620

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	#9649
prime radical	C₂²×#135	C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 497 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=df,c^2=e,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=fg,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, 47816);