Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #1082705 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C210 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C22×#537 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
5256 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,4)(6,8)(13,14,18,16)(15,20,23,19)(17,21,26,24)(22,28,25,27);(10,12)(13,15,18,23)(14,19,16,20)(17,25,26,22)(21,28,24,27);(1,2)(3,4)(5,6)(7,8)(9,10,11,12)(13,15)(14,20)(16,19)(17,22)(18,23)(21,28)(24,27)(25,26);(5,6)(7,8)(13,16,18,14)(15,20,23,19)(17,24,26,21)(22,28,25,27);(1,2)(3,4)(5,6)(7,8)(13,17)(14,21)(15,22)(16,24)(18,26)(19,27)(20,28)(23,25)Polycyclic presentation:
2^9|a^2=f,b^2=f,c^2=fh,d^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^b=ef,d^c=df86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 991936);