Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #1418353 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | #14715 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
4420 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2,4,3)(6,8)(9,10)(11,16)(12,14)(13,15)(17,18)(19,20)(21,22)(23,24);(2,3)(9,11)(10,14)(12,13)(15,16)(17,19,22,24)(18,23,21,20);(9,10)(11,16)(12,14)(13,15)(17,20)(18,24)(19,21)(22,23);(2,3)(5,6)(7,8)(9,10)(11,14)(12,16)(13,15)(17,20)(18,19)(21,24)(22,23);(1,3,4,2)(9,12,13,11)(10,14,15,16)(17,21,22,18)(19,23,24,20)Polycyclic presentation:
2^9|a^2=fh,b^2=g,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^b=ef,e^d=ef92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1326600);