Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2805613 |
| automorphism | Aut(G) order 4194304 |
| 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 | C22×#121 | C22 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3294 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2,4,3)(6,8)(13,14,18,21)(15,16,23,19)(17,25,28,20)(22,27,26,24);(2,3)(10,12)(13,15,18,23)(14,19,21,16)(17,26,28,22)(20,24,25,27);(9,10,11,12)(13,15)(14,19)(16,21)(17,22)(18,23)(20,27)(24,25)(26,28);(2,3)(5,6,7,8)(13,16,18,19)(14,15,21,23)(17,27,28,24)(20,26,25,22);(1,3,4,2)(13,17)(14,20)(15,22)(16,24)(18,28)(19,27)(21,25)(23,26)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=gh,d^2=fgi,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dg,e^b=ef,e^d=ef80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2830460);