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 | #53680 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4174 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)(3,11)(4,5)(6,9)(7,10)(8,14)(12,15)(13,16)(17,18)(19,22)(20,25)(21,26)(23,28)(24,30)(27,31)(29,32)(33,34,37,40)(35,41,43,38)(36,44,46,39)(42,47,45,48);(1,3,7,14)(2,8,10,11)(4,12,9,16)(5,13,6,15)(17,19,22,18)(20,25,28,23)(21,24,30,26)(27,31,29,32)(33,35,37,43)(34,38,40,41)(36,45,46,42)(39,48,44,47);(1,4,7,9)(2,5,10,6)(3,12,14,16)(8,13,11,15)(17,20,22,28)(18,23,19,25)(21,29,30,27)(24,32,26,31)(33,36,37,46)(34,39,40,44)(35,42,43,45)(38,47,41,48);(1,5,7,6)(2,9,10,4)(3,13,14,15)(8,16,11,12)(17,21,22,30)(18,24,19,26)(20,27,28,29)(23,31,25,32)(34,40)(35,43)(39,44)(42,45);(3,14)(8,11)(12,16)(13,15)(18,19)(23,25)(24,26)(31,32)(34,40)(38,41)(39,44)(47,48);(1,6,7,5)(2,9,10,4)(3,15,14,13)(8,16,11,12)Polycyclic presentation:
2^9|a^2=i,b^2=g,c^2=g,d^2=gi,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, 8011646);