#143599

order	512
nilpotency class	4
isoclinism	isoclinic class of #143516
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1999
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	#466
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₈×Q₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

322 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 322 conjugacy classes of subgroups — too many to draw. Only its 64 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) = 256

A minimal faithful permutation representation — 3 generators on 64 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=g,d^2=gi,e^2=hi,f^2=hi,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ehi,f^a=fh,g^a=gi,c^b=cf,d^b=dg,f^b=fhi,g^b=gi,d^c=dh,e^c=eh,f^c=fh

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 50795);