Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D4×D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| socle | C23 | C23×D4 |
| prime radical | C22×#135 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C4×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2190 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)(14,19)(15,22)(16,17)(20,27)(21,25)(26,28);(13,14)(15,20)(16,19)(17,25)(18,21)(22,28)(23,26)(24,27);(6,9)(10,12)(13,15,18,22)(14,20,21,28)(16,23,17,24)(19,26,25,27);(13,16)(15,23)(17,18)(19,25)(22,24)(26,27);(5,7,8,11)(6,10,9,12)Polycyclic presentation:
2^9|a^2=h,c^2=i,e^2=gi,f^2=i,b^a=bf,c^a=cg,d^a=di,f^a=fi,d^b=df,f^b=fi,f^d=fi140 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6288073);