#141317

order	512
nilpotency class	4
isoclinism	isoclinic class of #141316
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#9747
commutator	C₄×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	#472
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

270 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 270 conjugacy classes of subgroups — too many to draw. Only its 84 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=df,c^2=eghi,d^2=ghi,e^2=gi,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,f^b=fhi,h^b=hi,d^c=di

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, 48513);