#141300

order	512
nilpotency class	4
isoclinism	isoclinic class of #141300
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#9743
commutator	C₄×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	#3147	C₂²
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

337 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 337 conjugacy classes of subgroups — too many to draw. Only its 92 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfgh,c^2=e,d^2=gh,e^2=g,f^2=h,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eg,g^a=gi,c^b=cg,d^b=dh,e^b=ei,f^b=fh,h^b=hi,f^c=fi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 48496);