Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C26 |
| prime radical | C24 | C25 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3653 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 64 points:
(1,2,6,7)(3,4,9,12)(5,8,13,14)(10,11,15,16)(18,22)(19,25)(20,28)(30,32)(33,34)(35,41)(36,37)(38,52)(39,40)(42,50)(43,53)(44,56)(45,48)(46,60)(47,54)(49,57)(51,63)(55,61)(58,59)(62,64);(2,7)(3,9)(8,14)(10,15)(17,18,21,22)(19,26,25,23)(20,24,28,29)(27,32,31,30)(34,40)(37,42)(41,54)(43,52)(45,56)(51,57)(55,60)(59,64);(1,3)(2,4)(5,10)(6,9)(7,12)(8,11)(13,15)(14,16)(17,18)(19,23)(20,24)(21,22)(25,26)(27,30)(28,29)(31,32)(33,35)(34,41)(36,48)(37,45)(38,46)(39,47)(40,54)(42,56)(43,55)(44,50)(49,62)(51,59)(52,60)(53,61)(57,64)(58,63);(1,4)(2,3)(5,11)(6,12)(7,9)(8,10)(13,16)(14,15)(17,19,21,25)(18,23,22,26)(20,27,28,31)(24,30,29,32)(33,36)(34,42)(35,44)(37,40)(38,49)(39,50)(41,45)(43,57)(46,58)(47,48)(51,52)(53,63)(54,56)(55,59)(60,64)(61,62);(1,5,6,13)(2,8,7,14)(3,10,9,15)(4,11,12,16)(17,20,21,28)(18,24,22,29)(19,27,25,31)(23,30,26,32)(33,37,39,42)(34,36,40,50)(35,45,47,56)(38,51,53,57)(41,48,54,44)(43,49,52,63)(46,59,61,64)(55,62,60,58);(33,38,39,53)(34,43,40,52)(35,46,47,61)(36,49,50,63)(37,51,42,57)(41,55,54,60)(44,58,48,62)(45,59,56,64)Polycyclic presentation:
2^9|a^2=hi,b^2=h,d^2=h,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^b=egi,d^c=dgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8022602);