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 | #2678 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4103 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,4)(3,9,10,7)(5,8,13,12)(11,15,16,14)(17,18,21,24)(19,25,27,22)(20,28,29,23)(26,30,32,31)(33,34,37,40)(35,38,41,44)(36,39,43,46)(42,45,47,48);(1,3,6,10)(2,7,4,9)(5,11,13,16)(8,14,12,15)(17,19,21,27)(18,22,24,25)(20,26,29,32)(23,30,28,31)(33,35)(34,38)(36,42)(37,41)(39,45)(40,44)(43,47)(46,48);(1,4,6,2)(3,9,10,7)(5,12,13,8)(11,15,16,14)(17,20,21,29)(18,23,24,28)(19,26,27,32)(22,30,25,31)(35,41)(36,43)(38,44)(39,46);(2,4)(7,9)(8,12)(14,15)(18,24)(19,27)(20,29)(30,31)(33,36,37,43)(34,39,40,46)(35,42,41,47)(38,45,44,48);(3,10)(7,9)(11,16)(14,15)(18,24)(19,27)(23,28)(26,32)(35,41)(38,44)(42,47)(45,48);(1,5,6,13)(2,8,4,12)(3,11,10,16)(7,14,9,15)Polycyclic presentation:
2^9|a^2=gi,b^2=g,c^2=g,d^2=i,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, 8011639);