Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #35346 |
| 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 | #35568 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
5051 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)(3,8)(4,6)(5,7)(9,10)(11,14)(12,13)(15,16);(1,3,5,4)(2,6,7,8)(17,18)(19,25)(20,23)(21,24)(22,27)(26,31)(28,30)(29,32);(10,13)(14,16)(17,19,21,27)(18,22,24,25)(20,28,29,26)(23,31,32,30);(17,20,21,29)(18,23,24,32)(19,26,27,28)(22,30,25,31);(1,4,5,3)(2,6,7,8)(18,24)(22,25)(23,32)(30,31);(9,11,12,15)(10,14,13,16)Polycyclic presentation:
2^9|b^2=g,c^2=i,d^2=i,e^2=g,f^2=h,b^a=bg,c^a=ch,c^b=ci,e^b=ei,d^c=di128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7536305);