#144885

order	512
nilpotency class	4
isoclinism	isoclinic class of #142394
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1613
commutator, Frattini, cosocle	C₂²×M₁₆	C₂³
prime radical	C₂⁴	AGL(1,ℤ/8)
LCS	G ▷ C₂²×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×M₁₆ ◁ G
LECS	G ▷ C₂²×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

416 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 416 conjugacy classes of subgroups — too many to draw. Only its 68 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) = 68
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 68 points:

(2,3)(5,6,11,17)(7,18,23,46)(8,24,30,62)(9,31,34,14)(10,35,38,47)(12,32,41,20)(13,28,45,27)(15,48,16,33)(19,54,57,68)(21,59,22,40)(25,53,63,66)(26,52,29,50)(36,49,58,51)(37,44,64,43)(39,61,67,60)(42,55,65,56);(1,2)(3,4)(5,7,11,23)(6,12,17,41)(8,25,30,63)(9,32,34,20)(10,36,38,58)(13,42,45,65)(14,46,31,18)(15,49,16,51)(19,27,57,28)(21,48,22,33)(24,39,62,67)(26,61,29,60)(35,40,47,59)(37,55,64,56)(43,54,44,68)(50,66,52,53);(1,3,4,2)(5,8,9,26,11,30,34,29)(6,13,14,43,17,45,31,44)(7,19,20,55,23,57,32,56)(10,37,33,28,38,64,48,27)(12,39,18,53,41,67,46,66)(15,50,35,62,16,52,47,24)(21,60,36,63,22,61,58,25)(40,54,49,42,59,68,51,65)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=d,d^2=i,e^2=gi,f^2=g,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=eg,f^a=fhi,g^a=gi,h^a=hi,c^b=cf,d^b=di,e^b=eh,f^b=fgi,g^b=gi,h^b=hi,e^c=egi,f^c=fg,g^c=gi,f^e=fi

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 52081);