#138662

order	512
nilpotency class	4
isoclinism	isoclinic class of #138660
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:

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

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,e^a=ei,g^a=gi,d^b=dh,e^b=eh,f^b=fh,h^b=hi,d^c=dh,e^c=ehi,f^c=fh,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, 45858);