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 | C22×#41 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4848 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 24 points:
(2,7)(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)(3,8)(4,5)(6,7)(9,10)(11,14)(12,13)(15,16);(1,3,6,4)(2,8,7,5)(9,11,12,15)(10,14,13,16);(1,4,6,3)(2,8,7,5);(1,5,6,8)(2,4,7,3)Polycyclic presentation:
2^9|a^2=g,b^2=g,d^2=h,e^2=hi,f^2=hi,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, 7572837);