Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #46350 |
| 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 | #2675 | C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1852 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)(6,9)(7,11)(13,14,18,15,17,20,27,22)(16,19,26,21,25,23,28,24);(5,6)(7,10)(8,9)(11,12)(13,14,18,15,17,20,27,22)(16,23,26,24,25,19,28,21);(2,4)(13,15,18,20,17,22,27,14)(16,24,26,19,25,21,28,23);(6,9)(10,12)(13,16)(14,19)(15,21)(17,25)(18,26)(20,23)(22,24)(27,28);(5,7,8,11)(6,10,9,12)(16,25)(19,23)(21,24)(26,28)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=h,e^2=f,h^2=i,b^a=bf,c^a=cg,e^a=ef,c^b=ci,d^b=dfi,d^c=di,e^d=ei104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6302563);