Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6629032 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#121 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#3038 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2348 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2)(3,6)(4,5)(7,8)(10,16)(11,18)(12,20)(13,14)(19,23)(21,22);(2,5)(6,8)(9,10,14,11,15,18,13,16)(12,17,20,19,22,24,21,23);(1,3,4,7)(2,6,5,8)(9,11,13,10,15,16,14,18)(12,19,21,17,22,23,20,24);(9,12)(10,17)(11,19)(13,21)(14,20)(15,22)(16,23)(18,24);(10,18)(11,16)(17,24)(19,23)Polycyclic presentation:
2^9|b^2=g,c^2=f,f^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dh,f^a=fi,g^a=gi,h^a=hi,e^b=ei,e^c=ei,h^d=hi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6536730);