#140278

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	#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

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

normal subgroups only

This group has 244 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)(3,13,17,36)(4,19,23,35)(5,24)(6,27)(7,11)(8,16,30,25)(9,22,34,56)(10,14)(12,28)(15,43)(18,50,49,39)(20,38)(21,33,44,60)(26,51)(29,55)(31,40,59,48)(32,58)(37,42)(41,52,53,57)(45,62)(46,64)(47,63)(54,61);(1,3,14,40,28,49,5,25,7,17,42,48,6,18,26,16)(2,8,24,50,27,59,10,36,11,30,51,39,12,31,37,13)(4,20,53,58,56,15,44,54,23,46,41,45,22,47,21,55)(9,32,52,64,19,29,60,43,34,62,57,38,35,61,33,63);(1,4,5,21,6,22,14,41,7,23,26,44,28,56,42,53)(2,9,10,33,12,35,24,57,11,34,37,60,27,19,51,52)(3,15,16,45,18,20,48,54,17,47,25,58,49,46,40,55)(8,29,13,38,31,32,39,43,30,61,36,64,59,62,50,63)

Polycyclic presentation:

2^9|a^2=f,b^2=dfg,c^2=d,d^2=g,e^2=hi,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, 47474);