Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C26 |
| prime radical | C22×2−1+4 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
5544 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 32 points:
(3,7)(5,8)(10,13)(11,15)(17,18,23,25)(19,24,27,22)(20,21,26,32)(28,30,31,29);(1,2,4,6)(3,5,7,8)(9,10)(11,14)(12,13)(15,16)(17,19,23,27)(18,24,25,22)(20,28,26,31)(21,30,32,29);(1,3)(2,5)(4,7)(6,8)(9,11,12,15)(10,14,13,16)(18,25)(19,27)(20,26)(29,30);(17,20)(18,21)(19,28)(22,29)(23,26)(24,30)(25,32)(27,31);(17,21)(18,26)(19,29)(20,25)(22,31)(23,32)(24,28)(27,30);(17,22,23,24)(18,27,25,19)(20,29,26,30)(21,31,32,28)Polycyclic presentation:
2^9|a^2=i,b^2=gh,c^2=g,f^2=i,b^a=bg,c^a=ch,f^a=fi,c^b=ci,e^b=ei,d^c=di,e^d=ei128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7538440);