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 | |
3424 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)(13,14,16,21,17,19,24,18)(15,20,23,27,22,26,28,25);(1,2,3,4)(6,9)(10,11)(14,18)(16,24)(19,21)(20,25)(23,28)(26,27);(5,7,8,12)(6,10,9,11)(14,19)(18,21)(20,26)(25,27);(13,15)(14,20)(16,23)(17,22)(18,25)(19,26)(21,27)(24,28);(15,22)(20,26)(23,28)(25,27)Polycyclic presentation:
2^9|a^2=h,b^2=fgh,c^2=gi,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, 6279391);