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 | C23×D4 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
5232 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)(3,7)(4,6)(5,8)(10,13)(11,15)(17,18,20,22)(19,23,24,21);(10,13)(14,16)(17,19,20,24)(18,21,22,23);(1,2,4,6)(3,5,8,7)(9,10,12,13)(11,14,15,16);(9,11)(10,14)(12,15)(13,16);(1,2)(3,7)(4,6)(5,8);(1,3)(2,5)(4,8)(6,7)Polycyclic presentation:
2^9|a^2=g,b^2=g,c^2=h,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, 7571800);