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 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
5044 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 32 points:
(1,2,4,6)(3,7,8,5)(9,10)(11,16)(12,15)(13,14)(17,18,22,20)(19,25,26,23)(21,24,29,28)(27,31,32,30);(1,2)(3,5)(4,6)(7,8)(10,14)(11,12)(17,19)(18,23)(20,25)(21,27)(22,26)(24,30)(28,31)(29,32);(1,3,4,8)(2,5,6,7)(9,11,13,12)(10,15,14,16)(17,20,22,18)(19,25,26,23)(21,28,29,24)(27,31,32,30);(2,6)(3,8)(9,12,13,11)(10,16,14,15)(18,20)(23,25)(24,28)(30,31);(2,6)(5,7)(10,14)(15,16)(19,26)(23,25)(27,32)(30,31);(17,21,22,29)(18,24,20,28)(19,27,26,32)(23,30,25,31)Polycyclic presentation:
2^9|a^2=gi,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, 8011637);