Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #35680 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C25 |
| commutator, Frattini, socle, cosocle | C23 | C26 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
5232 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 24 points:
(1,2,5,8)(3,7,4,6)(9,10,13,16)(11,15,12,14)(17,18,23,22)(19,20,21,24);(1,3,5,4)(2,6,8,7)(9,11,13,12)(10,14,16,15)(17,19,23,21)(18,24,22,20);(1,3,5,4)(2,7,8,6)(9,12,13,11)(10,15,16,14)(17,20,23,24)(18,19,22,21);(17,21,23,19)(18,24,22,20);(17,22,23,18)(19,20,21,24);(1,4,5,3)(2,6,8,7)Polycyclic presentation:
2^9|a^2=g,b^2=g,c^2=g,d^2=i,e^2=i,f^2=gh,b^a=bg,c^a=ch,c^b=ci,e^d=ei128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7536855);