Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6628851 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3180 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,4)(6,8)(9,10)(11,15)(12,18)(13,16)(14,22)(17,19)(20,23)(21,24);(1,2,3,4)(9,11)(10,15)(12,21)(13,20)(14,19)(16,23)(17,22)(18,24);(5,6,7,8)(10,16)(15,23)(17,24)(18,22);(9,11)(10,17)(12,19)(13,20)(14,21)(15,22)(16,24)(18,23);(9,12,13,14)(10,18,16,22)(11,19,20,21)(15,17,23,24)Polycyclic presentation:
2^9|b^2=f,c^2=gi,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dh,h^a=hi,e^b=ei,h^d=hi98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6537206);