#144915

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

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

normal subgroups only

This group has 386 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)(5,6,12,19)(7,13,24,43)(8,25,27,60)(9,31,36,49)(10,18,11,17)(14,29,45,28)(15,35,16,34)(20,44,53,68)(21,50,58,47)(22,42,23,41)(26,61,54,38)(30,48,37,51)(32,40,57,39)(33,46,62,52)(55,65,59,66)(56,64,63,67);(1,2)(3,4)(5,7,12,24)(6,13,19,43)(8,26,27,54)(9,32,36,57)(10,23,11,22)(14,44,45,68)(15,47,16,50)(17,42,18,41)(20,28,53,29)(21,34,58,35)(25,38,60,61)(30,63,37,56)(31,40,49,39)(33,59,62,55)(46,65,52,66)(48,64,51,67);(1,3,4,2)(5,8,10,37,12,27,11,30)(6,14,17,52,19,45,18,46)(7,20,22,59,24,53,23,55)(9,33,34,29,36,62,35,28)(13,38,41,67,43,61,42,64)(15,48,31,60,16,51,49,25)(21,56,57,26,58,63,32,54)(39,65,50,68,40,66,47,44)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=dg,e^2=gi,f^2=g,g^2=i,h^2=i,b^a=bd,c^a=ce,e^a=eg,f^a=fh,g^a=gi,h^a=hi,c^b=cf,e^b=eh,f^b=fgi,g^b=gi,h^b=hi,d^c=di,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, 52111);