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 | |
3703 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 64 points:
(2,6)(7,12)(8,13)(14,16)(17,18,23,26)(19,27,30,21)(20,25,22,24)(28,31,29,32)(33,34)(35,41)(36,42)(37,43)(38,44)(39,40)(45,47)(46,56)(48,57)(49,54)(50,58)(51,52)(53,55)(59,60)(61,63)(62,64);(4,11)(8,13)(9,15)(14,16)(17,19,23,30)(18,21,26,27)(20,28,22,29)(24,31,25,32)(34,40)(37,41)(42,54)(44,55)(47,50)(51,56)(57,64)(60,63);(1,2)(3,7)(4,8)(5,6)(9,14)(10,12)(11,13)(15,16)(17,19)(18,21)(20,28)(22,29)(23,30)(24,31)(25,32)(26,27)(33,35,39,43)(34,41,40,37)(36,45,49,58)(38,52,53,46)(42,47,54,50)(44,51,55,56)(48,61,62,59)(57,63,64,60);(1,3,5,10)(2,7,6,12)(4,9,11,15)(8,14,13,16)(17,20,23,22)(18,24,26,25)(19,28,30,29)(21,31,27,32)(33,36)(34,42)(35,45)(37,50)(38,48)(39,49)(40,54)(41,47)(43,58)(44,57)(46,59)(51,63)(52,61)(53,62)(55,64)(56,60);(1,4,5,11)(2,8,6,13)(3,9,10,15)(7,14,12,16)(17,21)(18,19)(20,31)(22,32)(23,27)(24,28)(25,29)(26,30)(33,37,39,41)(34,43,40,35)(36,47,49,50)(38,51,53,56)(42,45,54,58)(44,52,55,46)(48,60,62,63)(57,59,64,61);(1,2)(3,7)(4,8)(5,6)(9,14)(10,12)(11,13)(15,16)(17,22,23,20)(18,25,26,24)(19,29,30,28)(21,32,27,31)(33,38,39,53)(34,44,40,55)(35,46,43,52)(36,48,49,62)(37,51,41,56)(42,57,54,64)(45,59,58,61)(47,60,50,63)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=gi,d^2=h,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,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, 8833913);