#140279

order	512
nilpotency class	4
isoclinism	isoclinic class of #140278
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#9571
commutator	C₄×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	#471
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

238 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 238 conjugacy classes of subgroups — too many to draw. Only its 78 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:

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

Polycyclic presentation:

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

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, 47475);