Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #1393169 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C210 |
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 | |
5400 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)(13,14,18,21)(15,22,16,19)(17,20,26,25)(23,28,24,27);(10,12)(13,15,18,16)(14,19,21,22)(17,23,26,24)(20,27,25,28);(1,2)(3,4)(5,6)(7,8)(9,10)(11,12)(13,16,18,15)(14,19,21,22)(17,24,26,23)(20,27,25,28);(5,6)(7,8)(13,14)(15,19)(16,22)(17,25)(18,21)(20,26)(23,28)(24,27);(1,2)(3,4)(5,6)(7,8)(13,17)(14,20)(15,23)(16,24)(18,26)(19,27)(21,25)(22,28)Polycyclic presentation:
2^9|a^2=f,b^2=f,c^2=f,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, 1300372);