Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #1393169 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C2×#2619 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3528 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,3)(6,8)(14,18)(15,21)(16,24)(27,28);(10,12)(13,14,17,18)(15,19,21,22)(16,20,24,26)(23,27,25,28);(1,2,4,3)(9,10)(11,12)(13,14)(15,22)(16,20)(17,18)(19,21)(23,28)(24,26)(25,27);(5,6)(7,8)(13,15)(14,19)(16,25)(17,21)(18,22)(20,28)(23,24)(26,27);(1,3,4,2)(13,16,17,24)(14,20,18,26)(15,23,21,25)(19,27,22,28)Polycyclic presentation:
2^9|b^2=f,c^2=fg,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^c=df,e^d=ef86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1300651);