Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #53390 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C25 |
| commutator, Frattini, socle, cosocle | C23 | C26 |
| prime radical | #52278 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4084 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,7,9)(3,10,13,8)(4,5,15,6)(11,14,16,12)(17,18)(19,22)(20,25)(21,26)(23,28)(24,30)(27,31)(29,32)(33,34,37,40)(35,38,42,47)(36,43,45,39)(41,48,44,46);(1,3)(2,8)(4,11)(5,12)(6,14)(7,13)(9,10)(15,16)(17,19,22,18)(20,25,28,23)(21,24,30,26)(27,31,29,32)(33,35,37,42)(34,38,40,47)(36,44,45,41)(39,48,43,46);(1,4)(2,5)(3,11)(6,9)(7,15)(8,12)(10,14)(13,16)(17,20,22,28)(18,23,19,25)(21,29,30,27)(24,32,26,31)(33,36,37,45)(34,39,40,43)(35,41,42,44)(38,46,47,48);(1,5,7,6)(2,4,9,15)(3,12,13,14)(8,11,10,16)(17,21,22,30)(18,24,19,26)(20,27,28,29)(23,31,25,32)(35,42)(36,45)(38,47)(39,43);(3,13)(8,10)(11,16)(12,14)(18,19)(23,25)(24,26)(31,32)(34,40)(35,42)(39,43)(41,44);(1,6,7,5)(2,4,9,15)(3,14,13,12)(8,11,10,16)Polycyclic presentation:
2^9|a^2=gh,b^2=h,c^2=h,d^2=g,f^2=ghi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,d^c=dgi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8012788);