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 | |
4180 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(2,7)(3,9)(4,11)(12,16)(20,28)(24,31)(27,32)(29,30)(33,34)(35,40)(36,47)(37,42)(38,43)(39,44)(41,46)(45,48);(1,2,6,7)(3,5,9,13)(4,8,11,14)(10,12,15,16)(17,18)(19,23)(20,24)(21,25)(22,26)(27,30)(28,31)(29,32)(33,35,39,43)(34,40,44,38)(36,37,46,45)(41,48,47,42);(17,19)(18,21)(20,27)(22,25)(23,26)(24,29)(28,32)(30,31)(33,36,39,46)(34,41,44,47)(35,45,43,37)(38,48,40,42);(1,3)(2,5)(4,10)(6,9)(7,13)(8,12)(11,15)(14,16)(17,19)(18,23)(20,27)(21,26)(22,25)(24,30)(28,32)(29,31)(33,35)(34,40)(36,45)(37,46)(38,44)(39,43)(41,42)(47,48);(1,4)(2,8)(3,10)(5,12)(6,11)(7,14)(9,15)(13,16)(17,20,22,28)(18,24,26,31)(19,27,25,32)(21,29,23,30)(33,37,39,45)(34,42,44,48)(35,46,43,36)(38,47,40,41);(1,5)(2,9)(3,7)(4,12)(6,13)(8,15)(10,14)(11,16)(17,21,22,23)(18,25,26,19)(20,29,28,30)(24,32,31,27)(33,38,39,40)(34,43,44,35)(36,48,46,42)(37,47,45,41)Polycyclic presentation:
2^9|b^2=gh,c^2=h,e^2=i,f^2=i,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,f^d=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8365655);