#142913

order	512
nilpotency class	4
isoclinism	isoclinic class of #142406
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1616
commutator, Frattini, cosocle	C₂²×M₁₆	C₂³
prime radical	C₂⁴	C₂×SD₁₆
LCS	G ▷ C₂²×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×M₁₆ ◁ G
LECS	G ▷ C₂²×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 414 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) = 68
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=i,d^2=gi,f^2=gi,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,f^a=fhi,g^a=gi,h^a=hi,c^b=cf,d^b=dgi,e^b=ei,f^b=fgi,g^b=gi,d^c=dh,f^c=fg,g^c=gi,h^c=hi,f^d=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, 50109);