Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #60372 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3554 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,4)(5,6)(7,11)(8,9)(10,12)(13,14,17,19,18,22,16,21)(15,20,25,26,23,27,24,28);(1,2,3,4)(6,9)(10,11)(14,19)(16,17)(20,26)(21,22)(24,25)(27,28);(5,7,8,12)(6,10,9,11);(13,15)(14,20)(16,24)(17,25)(18,23)(19,26)(21,28)(22,27);(15,23)(20,27)(24,25)(26,28)Polycyclic presentation:
2^9|a^2=h,b^2=fgh,c^2=g,f^2=i,h^2=i,b^a=bf,c^a=cg,f^b=fi,h^b=hi,e^d=ei122 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6279389);