Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #37644 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C4 | C25 |
| commutator, socle | C23 | C24×C4 |
| prime radical | #34925 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2349 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)(3,4)(5,6)(7,10)(8,9)(11,12)(13,14,19,17,18,22,16,21)(15,23,27,28,24,20,25,26);(6,9)(7,11)(13,15)(14,20)(16,25)(17,26)(18,24)(19,27)(21,28)(22,23);(2,4)(15,24)(20,23)(25,27)(26,28);(5,7,8,11)(6,10,9,12)(13,16,18,19)(14,21,22,17)(15,25,24,27)(20,28,23,26);(6,9)(10,12)(13,17,19,22,18,21,16,14)(15,26,27,23,24,28,25,20)Polycyclic presentation:
2^9|a^2=h,d^2=f,e^2=h,h^2=i,b^a=bf,c^a=cg,e^a=ef,c^b=ci,d^b=dfi116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6306618);