#144514

order	512
nilpotency class	4
isoclinism	isoclinic class of #144514
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#10348
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	#3147	C₂²
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

319 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 319 conjugacy classes of subgroups — too many to draw. Only its 78 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)(3,8)(4,18,20,9)(5,23)(6,10)(7,29)(11,39)(12,28)(13,45,46,62)(14,32)(15,50)(16,43)(17,34)(19,35,55,53)(21,58,57,41)(22,37,30,44)(24,60)(25,54,59,36)(26,42)(27,38)(31,47,61,48)(33,49)(40,52)(51,64,56,63);(1,3)(2,8)(4,19,20,55)(5,24)(6,17)(7,14)(9,35,18,53)(10,34)(11,40)(12,43)(13,21,46,57)(15,27)(16,28)(22,51,30,56)(23,60)(25,47,59,48)(26,49)(29,32)(31,36,61,54)(33,42)(37,63,44,64)(38,50)(39,52)(41,62,58,45);(1,4)(2,9)(3,13)(5,25)(6,20)(7,30)(8,31)(10,18)(11,41)(12,44)(14,48)(15,51)(16,47)(17,46)(19,49)(21,39)(22,28)(23,36)(24,55)(26,59)(27,57)(29,37)(32,62)(33,63)(34,61)(35,50)(38,58)(40,53)(42,54)(43,45)(52,56)(60,64)

Polycyclic presentation:

2^9|a^2=d,b^2=d,e^2=gi,f^2=h,h^2=i,b^a=bd,c^a=ce,e^a=eg,f^a=fhi,h^a=hi,c^b=cf,e^b=eh,f^b=fh,g^b=gi,h^b=hi,d^c=di,e^c=egi,f^c=fhi,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, 51710);