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 | C2×#534 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
5706 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 24 points:
(1,2,6,4)(3,8,5,7)(9,10,12,15)(11,14,16,13)(17,18,20,21)(19,23,24,22);(9,11)(10,13)(12,16)(14,15)(18,21)(22,23);(1,3,6,5)(2,7,4,8)(9,10)(11,13)(12,15)(14,16)(17,18)(19,22)(20,21)(23,24);(17,19)(18,22)(20,24)(21,23);(1,4,6,2)(3,7,5,8)(9,11)(10,14)(12,16)(13,15);(1,5,6,3)(2,8,4,7)(9,10)(11,13)(12,15)(14,16)Polycyclic presentation:
2^9|a^2=h,c^2=gh,e^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=di,f^a=fhi,c^b=ci,e^c=ehi,f^e=fhi110 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7572052);