Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #16249 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C23×C4 | C24 |
| commutator, socle | C24 | C23×C4 |
| prime radical | C25 | C22×C4 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1250 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 28 points:
(5,6)(7,10)(8,11)(9,12)(13,14)(15,19)(16,18)(17,20)(21,22,24,26,23,25,27,28);(1,2)(3,4)(5,7,10,6)(8,12,9,11)(13,15)(14,17)(16,20)(18,19)(22,25)(26,28);(2,4)(5,8,10,9)(6,11,7,12)(14,18)(17,19)(22,25)(26,28);(5,9,10,8)(6,11,7,12)(15,20)(17,19)(22,25)(26,28)Polycyclic presentation:
2^9|a^2=h,b^2=ef,c^2=ef,d^2=ef,h^2=i,b^a=be,c^a=cf,d^a=defi,c^b=cg,d^b=dei104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 293009);