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 | #37272 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4820 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)(3,4)(5,6)(7,8)(10,13)(14,16)(17,18,22,24)(19,27,28,23)(20,25,30,32)(21,31,29,26);(9,10,12,13)(11,14,15,16)(17,19,22,28)(18,23,24,27)(20,29,30,21)(25,31,32,26);(2,6)(3,7)(18,24)(19,28)(20,30)(26,31);(1,3,5,7)(2,4,6,8)(9,11,12,15)(10,14,13,16)(17,20)(18,25)(19,29)(21,28)(22,30)(23,31)(24,32)(26,27);(1,4)(2,7)(3,6)(5,8)(17,18)(19,23)(20,25)(21,26)(22,24)(27,28)(29,31)(30,32);(17,21,22,29)(18,26,24,31)(19,20,28,30)(23,25,27,32)Polycyclic presentation:
2^9|a^2=i,b^2=g,d^2=gh,f^2=i,b^a=bg,c^a=ch,e^a=eh,f^a=fi,c^b=ci,d^c=dh,e^c=ei104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7619338);