#140242

order	512
nilpotency class	4
isoclinism	isoclinic class of #12465
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1677
commutator	C₄×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂	#9564
prime radical	#3147	C₂²
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

248 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 248 conjugacy classes of subgroups — too many to draw. Only its 82 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfg,c^2=dh,d^2=gi,e^2=hi,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gi,e^b=ei,d^c=di,e^c=eghi,f^c=fi,h^c=hi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 47438);