Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #40484 |
| 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 | |
4844 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 32 points:
(1,2,5,8)(3,7,4,6)(17,18,22,25)(19,21,26,23)(20,28,30,24)(27,29,32,31);(1,3,5,4)(2,6,8,7)(9,10)(11,15)(12,14)(13,16)(17,19,22,26)(18,23,25,21)(20,27,30,32)(24,29,28,31);(1,3,5,4)(2,7,8,6)(9,11)(10,13)(12,16)(14,15)(17,20,22,30)(18,24,25,28)(19,27,26,32)(21,31,23,29);(10,14)(13,15)(18,25)(19,26)(24,28)(27,32);(1,3,5,4)(2,7,8,6)(11,16)(13,15)(17,21,22,23)(18,26,25,19)(20,29,30,31)(24,27,28,32);(1,4,5,3)(2,6,8,7)Polycyclic presentation:
2^9|a^2=g,b^2=g,c^2=g,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dh,c^b=ci,d^b=dhi,e^c=ehi104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7546818);