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 | C2×#121 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3709 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)(3,9)(4,11)(5,13)(6,8)(7,14)(10,12)(15,16)(18,23)(21,29)(24,30)(28,32)(33,34)(35,41)(36,42)(37,43)(38,44)(39,40)(45,48)(46,57)(47,51)(49,58)(50,55)(52,53)(54,56)(59,61)(60,63)(62,64);(1,2,6,8)(3,9,11,4)(5,7,14,13)(10,15,16,12)(20,27)(24,30)(26,31)(28,32)(34,40)(37,41)(42,55)(44,56)(48,51)(52,57)(58,64)(61,63);(1,3)(2,4)(5,10)(6,11)(7,12)(8,9)(13,15)(14,16)(17,18,22,23)(19,21,25,29)(20,24,27,30)(26,28,31,32)(33,35,39,43)(34,41,40,37)(36,47,50,45)(38,53,54,46)(42,51,55,48)(44,52,56,57)(49,59,62,60)(58,61,64,63);(1,4)(2,3)(5,12)(6,9)(7,10)(8,11)(13,16)(14,15)(17,19,22,25)(18,21,23,29)(20,26,27,31)(24,28,30,32)(33,36,39,50)(34,42,40,55)(35,45,43,47)(37,51,41,48)(38,49,54,62)(44,58,56,64)(46,59,53,60)(52,63,57,61);(1,2)(3,4)(5,7)(6,8)(9,11)(10,12)(13,14)(15,16)(17,20)(18,24)(19,26)(21,28)(22,27)(23,30)(25,31)(29,32)(33,37,39,41)(34,43,40,35)(36,48,50,51)(38,52,54,57)(42,45,55,47)(44,53,56,46)(49,61,62,63)(58,59,64,60);(1,5,6,14)(2,7,8,13)(3,10,11,16)(4,12,9,15)(17,21)(18,25)(19,23)(20,28)(22,29)(24,31)(26,30)(27,32)(33,38,39,54)(34,44,40,56)(35,46,43,53)(36,49,50,62)(37,52,41,57)(42,58,55,64)(45,59,47,60)(48,61,51,63)Polycyclic presentation:
2^9|b^2=i,c^2=gh,d^2=gh,e^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dgi,f^c=fgi,e^d=egi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8839286);