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 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
5784 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)(5,6)(7,8)(10,13)(14,15);(1,3)(2,4)(5,8)(6,7)(9,10)(11,15)(12,13)(14,16)(17,18)(19,24)(20,21)(22,23);(2,6)(3,8)(9,10,12,13)(11,14,16,15)(18,23)(19,20);(17,19,22,20)(18,24,23,21);(1,3)(2,7)(4,6)(5,8)(9,10)(11,14)(12,13)(15,16)(17,20,22,19)(18,21,23,24);(1,4,5,7)(2,3,6,8)(9,11,12,16)(10,14,13,15)(17,21)(18,19)(20,23)(22,24)Polycyclic presentation:
2^9|c^2=g,d^2=hi,e^2=hi,f^2=h,b^a=bg,c^a=ch,e^a=eh,c^b=ci,f^b=fi,d^c=dhi,e^c=ei104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7644832);