#140288

order	512
nilpotency class	4
isoclinism	isoclinic class of #140287
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#9572
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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfg,c^2=dh,d^2=g,e^2=h,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,f^b=fi,d^c=di,e^c=egh,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, 47484);