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 | |
5295 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 24 points:
(9,10,12,14)(11,13,15,16)(17,18)(19,23)(20,21)(22,24);(1,2,5,6)(3,7,8,4)(9,10)(11,13)(12,14)(15,16);(2,6)(4,7)(18,21)(22,23);(1,3,5,8)(2,7,6,4)(17,18)(19,22)(20,21)(23,24);(1,3)(2,4)(5,8)(6,7)(17,19)(18,22)(20,24)(21,23);(1,4)(2,3)(5,7)(6,8)(9,11,12,15)(10,13,14,16)Polycyclic presentation:
2^9|a^2=g,b^2=i,d^2=i,f^2=g,b^a=bg,c^a=ch,e^a=eh,c^b=ci,e^b=ei,d^c=dh,f^c=fi,e^d=ei98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7698278);