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 | C2×#121 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3990 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)(3,11)(4,8)(5,10)(6,9)(7,13)(12,14)(15,16)(17,18,22,25)(19,21,27,23)(20,28,30,24)(26,29,32,31)(33,34)(35,41)(36,39)(37,40)(38,43)(42,47)(44,46)(45,48);(1,3,6,13)(2,7,9,11)(4,12,10,16)(5,14,8,15)(17,19,22,27)(18,23,25,21)(20,26,30,32)(24,29,28,31)(33,35,37,43)(34,38,40,41)(36,44,45,42)(39,47,48,46);(1,4,6,10)(2,8,9,5)(3,12,13,16)(7,15,11,14)(17,20,22,30)(18,24,25,28)(19,26,27,32)(21,31,23,29)(33,36,37,45)(34,39,40,48)(35,42,43,44)(38,46,41,47);(2,9)(4,10)(7,11)(12,16)(17,21,22,23)(18,19,25,27)(20,29,30,31)(24,32,28,26)(34,40)(38,41)(39,48)(46,47);(3,13)(7,11)(12,16)(14,15)(18,25)(19,27)(24,28)(26,32);(1,5,6,8)(2,10,9,4)(3,14,13,15)(7,16,11,12)Polycyclic presentation:
2^9|a^2=h,b^2=g,c^2=g,d^2=h,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, 8012809);