#139137

order	512
nilpotency class	4
isoclinism	isoclinic class of #139080
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1879
commutator	C₄×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	#3169	C₂²
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1746 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

289 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 289 conjugacy classes of subgroups — too many to draw. Only its 72 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) = 128

A minimal faithful permutation representation — 3 generators on 64 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfhi,c^2=g,d^2=ghi,e^2=h,f^2=gh,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eg,d^b=dh,e^b=eghi,f^b=fh,h^b=hi,d^c=dghi,e^c=ehi,f^c=fghi,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, 46333);