Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #1393169 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C22×#121 | C22 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3622 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)(6,7)(13,14,18,22)(15,23,25,19)(16,26,27,20)(17,24,28,21);(10,12)(13,15,18,25)(14,19,22,23)(16,24,27,21)(17,20,28,26);(1,2,4,3)(5,6,8,7)(9,10,11,12)(13,15)(14,19)(16,21)(17,20)(18,25)(22,23)(24,27)(26,28);(5,6,8,7)(13,16,18,27)(14,20,22,26)(15,24,25,21)(17,19,28,23);(1,3,4,2)(5,7,8,6)(13,17,18,28)(14,21,22,24)(15,20,25,26)(16,23,27,19)Polycyclic presentation:
2^9|a^2=f,b^2=f,c^2=gh,d^2=i,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, 1316440);