#143595

order	512
nilpotency class	4
isoclinism	isoclinic class of #143518
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	#469
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2960 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

317 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 317 conjugacy classes of subgroups — too many to draw. Only its 66 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)(34,39)(35,44)(36,48)(37,50)(40,56)(41,49)(42,54)(43,55)(45,61)(46,63)(47,60)(51,52)(53,57)(58,62);(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)(33,34,38,43)(35,45,47,40)(36,49,51,54)(37,41,50,42)(39,52,55,48)(44,58,60,57)(46,56,63,61)(53,59,62,64);(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)(33,35,37,44,38,47,50,60)(34,40,42,58,43,45,41,57)(36,46,48,59,51,63,52,64)(39,53,49,61,55,62,54,56)

Polycyclic presentation:

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

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, 50791);