Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2735949 |
| automorphism | Aut(G) order 4194304 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
4398 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,4)(10,13)(11,15)(18,21)(19,23);(6,8)(9,10)(11,14)(12,13)(15,16)(19,23)(22,24);(5,6,7,8)(9,10,12,13)(11,14,15,16)(17,18)(19,22)(20,21)(23,24);(1,2)(3,4)(9,10)(11,16)(12,13)(14,15)(17,19)(18,22)(20,23)(21,24);(9,11)(10,14)(12,15)(13,16)(17,18)(19,22)(20,21)(23,24)Polycyclic presentation:
2^9|c^2=h,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,d^c=df,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2643729);