#140301

order	512
nilpotency class	4
isoclinism	isoclinic class of #140301
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#9576
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

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

normal subgroups only

This group has 201 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)(3,13,17,40)(4,19,24,39)(5,25)(6,29)(7,11)(8,16,33,26)(9,23,38,59)(10,28)(12,30)(14,36)(15,46)(18,51,50,43)(20,53)(21,31)(22,37,27,41)(32,58)(34,55,60,44)(35,49)(42,57)(45,52,54,61)(47,62)(48,64)(56,63);(1,3,14,44,30,50,21,16,7,17,28,55,6,18,5,26)(2,8,31,43,29,60,36,13,11,33,25,51,12,34,10,40)(4,20,54,49,59,15,27,56,24,57,45,47,23,48,22,58)(9,35,52,42,19,32,41,46,38,62,61,53,39,63,37,64);(1,4,21,22,6,23,14,45,7,24,5,27,30,59,28,54)(2,9,36,37,12,39,31,61,11,38,10,41,29,19,25,52)(3,15,26,47,18,20,55,56,17,48,16,49,50,57,44,58)(8,32,40,53,34,35,51,46,33,63,13,42,60,62,43,64)

Polycyclic presentation:

2^9|a^2=f,b^2=dfgh,c^2=dh,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,f^b=fi,d^c=di,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, 47497);