#144076

order	512
nilpotency class	4
isoclinism	isoclinic class of #144076
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#10278
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂×#135	D₄
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

299 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 299 conjugacy classes of subgroups — too many to draw. Only its 80 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)(3,18,24,49)(4,25)(5,14)(6,13)(7,16)(8,15)(9,17)(10,20,41,56)(11,28)(12,34)(19,51,54,59)(21,40,22,50)(23,38,36,39)(26,42,55,64)(27,46)(29,45)(30,48)(31,43)(32,60)(33,44)(35,47)(37,53,62,58)(52,61,57,63);(1,3)(2,10)(4,26)(5,22)(6,20)(7,36)(8,23)(9,24)(11,42)(12,38)(13,39)(14,49)(15,50)(16,40)(17,41)(18,47)(19,29)(21,35)(25,59)(27,58)(28,53)(30,57)(31,52)(32,55)(33,54)(34,56)(37,43)(44,61)(45,63)(46,64)(48,62)(51,60);(1,4)(2,11)(3,19,24,54)(5,33)(6,28)(7,31)(8,30)(9,32)(10,37,41,62)(12,43)(13,48)(14,25)(15,45)(16,44)(17,46)(18,42,49,64)(20,55,56,26)(21,52,22,57)(23,58,36,53)(27,34)(29,35)(38,63,39,61)(40,51,50,59)(47,60)

Polycyclic presentation:

2^9|a^2=gh,c^2=gh,d^2=g,e^2=hi,f^2=gi,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,f^a=fg,g^a=gi,h^a=hi,c^b=cf,d^b=dgi,e^b=ei,f^b=fg,g^b=gi,d^c=dg,e^c=eh,f^c=fgi,g^c=gi,h^c=hi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 51272);