Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #1258513 |
| 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 | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3150 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)(13,14,18,22)(15,20,16,19)(17,25,26,21)(23,27,24,28);(10,12)(13,15,18,16)(14,19,22,20)(17,23,26,24)(21,27,25,28);(1,2)(3,4)(9,10)(11,12)(13,16,18,15)(14,20,22,19)(17,24,26,23)(21,28,25,27);(5,6,7,8)(13,17,18,26)(14,21,22,25)(15,23,16,24)(19,27,20,28);(1,2)(3,4)(13,14)(15,19)(16,20)(17,25)(18,22)(21,26)(23,28)(24,27)Polycyclic presentation:
2^9|a^2=f,b^2=f,c^2=f,d^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^d=ef86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1200532);