Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4×#120 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C4 | C25 |
| commutator, socle | C23 | C24×C4 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2468 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)(13,14,19,15,18,22,16,20)(17,21,26,23,25,28,24,27);(6,9)(10,12)(13,15,19,22,18,20,16,14)(17,23,26,28,25,27,24,21);(2,4)(5,7,8,11)(6,10,9,12)(13,16,18,19)(14,20,22,15)(17,24,25,26)(21,27,28,23);(13,17)(14,21)(15,23)(16,24)(18,25)(19,26)(20,27)(22,28);(17,25)(21,28)(23,27)(24,26)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=f,h^2=i,b^a=bf,c^a=cg,e^d=ei140 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6278807);