#138665

order	512
nilpotency class	4
isoclinism	isoclinic class of #138663
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#9221
commutator	C₄×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂×#135	D₄
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

352 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 352 conjugacy classes of subgroups — too many to draw. Only its 94 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfghi,d^2=gh,e^2=h,f^2=hi,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gi,d^b=dh,e^b=ehi,f^b=fh,h^b=hi,d^c=dhi,e^c=ehi,f^c=fhi,h^c=hi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 45861);