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 | #58997 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
5535 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)(9,10)(11,15)(12,14)(13,16)(17,18,21,20)(19,23,24,22);(1,3,7,5)(2,4,8,6)(9,11)(10,13)(12,16)(14,15)(17,19,21,24)(18,22,20,23);(1,4,7,6)(2,5,8,3)(10,14)(13,15);(1,3,7,5)(2,4,8,6)(9,10)(11,13)(12,14)(15,16)(17,18,21,20)(19,22,24,23);(1,5,7,3)(2,4,8,6)(9,11)(10,13)(12,16)(14,15)(17,20,21,18)(19,23,24,22);(1,6,7,4)(2,5,8,3)Polycyclic presentation:
2^9|a^2=ghi,b^2=gh,c^2=i,d^2=gh,e^2=gh,f^2=i,b^a=bg,c^a=ch,e^a=eh,c^b=ci,d^c=dh,f^e=fi101 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7790231);