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 | |
4955 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)(17,18,23,25)(19,27,30,24)(20,21,26,22)(28,29,32,31);(1,3,4,8)(2,5,6,7)(10,13)(11,15)(17,19,23,30)(18,24,25,27)(20,28,26,32)(21,31,22,29);(2,6)(3,8)(9,10)(11,14)(12,13)(15,16)(17,20)(18,21)(19,28)(22,25)(23,26)(24,31)(27,29)(30,32);(2,6)(3,8)(10,13)(14,16)(18,25)(20,26)(24,27)(28,32);(2,6)(5,7)(9,11,12,15)(10,14,13,16)(17,21,23,22)(18,26,25,20)(19,29,30,31)(24,28,27,32);(17,22,23,21)(18,20,25,26)(19,29,30,31)(24,28,27,32)Polycyclic presentation:
2^9|a^2=g,b^2=g,e^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, 8012422);