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 | #39383 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
5170 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)(3,4)(5,8)(6,7)(10,14)(15,16)(17,18)(19,21)(20,23)(22,24);(1,3,6,5)(2,4,7,8)(9,10,13,14)(11,15,12,16)(18,19)(22,23);(2,7)(4,8)(9,11,13,12)(10,15,14,16)(17,19,21,18)(20,23,24,22);(17,20)(18,22)(19,23)(21,24);(1,4,6,8)(2,3,7,5)(9,11,13,12)(10,15,14,16);(1,5,6,3)(2,4,7,8)(9,12,13,11)(10,16,14,15)Polycyclic presentation:
2^9|b^2=gh,c^2=g,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, 7572015);