Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #40484 |
| 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 | #34624 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2086 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)(13,14,19,16,18,21,17,26)(15,22,24,28,25,20,27,23);(6,9)(7,11)(13,15,19,24,18,25,17,27)(14,20,16,23,21,22,26,28);(2,4)(13,16,19,21,18,26,17,14)(15,23,24,22,25,28,27,20);(5,7,8,11)(6,10,9,12)(13,17,18,19)(14,16,21,26)(15,24,25,27)(20,28,22,23);(6,9)(10,12)(14,21)(16,26)(20,22)(23,28)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=h,d^2=f,h^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ef,d^b=df,d^c=di,e^c=ei104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6302096);