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 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4012 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2)(3,5)(4,8)(6,7)(9,13)(10,12)(11,14)(15,16)(17,18)(19,28)(20,22)(21,24)(23,25)(26,27)(29,32)(30,31)(33,34,39,36)(35,37,40,42)(38,44,47,41)(43,46,48,45);(4,11)(8,14)(10,15)(12,16)(17,19,23,21)(18,24,25,28)(20,29,27,31)(22,30,26,32)(33,35)(34,37)(36,42)(38,43)(39,40)(41,45)(44,46)(47,48);(2,7)(3,9)(8,14)(10,15)(18,25)(20,27)(24,28)(29,31)(34,36)(35,40)(41,44)(43,48);(1,3,6,9)(2,5,7,13)(4,10,11,15)(8,12,14,16)(17,20,23,27)(18,26,25,22)(19,29,21,31)(24,32,28,30)(33,36,39,34)(35,42,40,37)(38,44,47,41)(43,46,48,45);(1,4)(2,8)(3,10)(5,12)(6,11)(7,14)(9,15)(13,16)(17,21,23,19)(18,24,25,28)(20,31,27,29)(22,30,26,32)(33,37,39,42)(34,40,36,35)(38,46,47,45)(41,43,44,48);(1,5)(2,9)(3,7)(4,12)(6,13)(8,15)(10,14)(11,16)(17,22,23,26)(18,27,25,20)(19,30,21,32)(24,31,28,29)(33,38)(34,41)(35,43)(36,44)(37,45)(39,47)(40,48)(42,46)Polycyclic presentation:
2^9|a^2=i,b^2=g,d^2=h,e^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9746540);