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 | #34589 | 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:
(1,2,7,4)(3,11,13,8)(5,9,6,10)(12,16,14,15)(17,18,22,25)(19,24,20,23)(21,29,30,26)(27,31,28,32)(33,34,37,40)(35,38,42,46)(36,39,44,48)(41,45,43,47);(1,3,7,13)(2,8,4,11)(5,14,6,12)(9,16,10,15)(17,19,22,20)(18,23,25,24)(21,28,30,27)(26,32,29,31)(33,35,37,42)(34,38,40,46)(36,43,44,41)(39,47,48,45);(1,4,7,2)(3,11,13,8)(5,10,6,9)(12,16,14,15)(17,20,22,19)(18,24,25,23)(21,28,30,27)(26,32,29,31)(33,36,37,44)(34,39,40,48)(35,41,42,43)(38,45,46,47);(2,4)(8,11)(9,10)(15,16)(18,25)(19,20)(26,29)(27,28)(36,44)(39,48)(41,43)(45,47);(1,5,7,6)(2,9,4,10)(3,12,13,14)(8,15,11,16)(17,21,22,30)(18,26,25,29)(19,27,20,28)(23,31,24,32)(35,42)(38,46)(41,43)(45,47);(1,6,7,5)(2,10,4,9)(3,12,13,14)(8,15,11,16)Polycyclic presentation:
2^9|a^2=gi,b^2=gi,c^2=gi,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, 8011125);