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 | #52280 | 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)(3,11)(4,9)(5,10)(6,8)(7,13)(12,16)(14,15)(17,18)(19,22)(20,25)(21,26)(23,27)(24,29)(28,30)(31,32)(33,34,37,40)(35,38,42,46)(36,39,44,48)(41,45,43,47);(1,3,6,13)(2,7,8,11)(4,14,5,12)(9,16,10,15)(17,19,22,18)(20,25,27,23)(21,24,29,26)(28,30,32,31)(33,35,37,42)(34,38,40,46)(36,43,44,41)(39,47,48,45);(1,2)(3,7)(4,9)(5,10)(6,8)(11,13)(12,15)(14,16)(17,20)(18,23)(19,25)(21,28)(22,27)(24,30)(26,31)(29,32)(33,36,37,44)(34,39,40,48)(35,41,42,43)(38,45,46,47);(2,8)(7,11)(9,10)(15,16)(18,19)(20,27)(24,26)(28,32)(36,44)(39,48)(41,43)(45,47);(1,4,6,5)(2,9,8,10)(3,12,13,14)(7,15,11,16)(17,21,22,29)(18,24,19,26)(20,28,27,32)(23,30,25,31)(35,42)(38,46)(41,43)(45,47);(1,5,6,4)(2,10,8,9)(3,12,13,14)(7,15,11,16)Polycyclic presentation:
2^9|a^2=i,b^2=gi,c^2=i,e^2=g,f^2=gh,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, 8011120);