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 | C22×D4 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3679 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)(3,7)(4,13)(5,9)(6,10)(8,12)(11,16)(14,15)(17,18,22,23)(19,20,24,27)(21,25,30,31)(26,28,32,29)(33,34)(35,39)(36,43)(37,41)(38,45)(40,48)(42,51)(44,49)(46,55)(47,53)(50,57)(52,58)(54,61)(56,60)(59,62)(63,64);(3,11)(5,14)(7,16)(9,15)(18,23)(20,27)(25,31)(28,29)(34,35)(37,49)(40,43)(42,45)(46,58)(50,61)(53,56)(59,64);(1,3,6,11)(2,7,10,16)(4,5,12,14)(8,15,13,9)(17,19,22,24)(18,20,23,27)(21,26,30,32)(25,28,31,29)(33,35,39,34)(36,40,48,43)(37,44,49,41)(38,45,51,42)(46,52,58,55)(47,53,60,56)(50,57,61,54)(59,62,64,63);(1,4,6,12)(2,8,10,13)(3,5,11,14)(7,15,16,9)(33,36,39,48)(34,40,35,43)(37,46,49,58)(38,47,51,60)(41,52,44,55)(42,53,45,56)(50,59,61,64)(54,62,57,63);(1,5)(2,9)(3,12)(4,11)(6,14)(7,8)(10,15)(13,16)(17,20)(18,24)(19,23)(21,29)(22,27)(25,26)(28,30)(31,32)(33,37,39,49)(34,41,35,44)(36,46,48,58)(38,50,51,61)(40,52,43,55)(42,54,45,57)(47,59,60,64)(53,62,56,63);(1,2)(3,7)(4,8)(5,15)(6,10)(9,14)(11,16)(12,13)(17,21)(18,25)(19,26)(20,28)(22,30)(23,31)(24,32)(27,29)(33,38,39,51)(34,42,35,45)(36,47,48,60)(37,50,49,61)(40,53,43,56)(41,54,44,57)(46,59,58,64)(52,62,55,63)Polycyclic presentation:
2^9|a^2=gh,c^2=ghi,d^2=i,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^e=fgi77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9831180);