Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6500482 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#121 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1903 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 24 points:
(10,14)(11,18)(12,15)(16,24)(17,22)(19,21);(1,2)(3,6)(4,5)(7,8)(9,10,15,22,13,17,12,14)(11,19,23,21,18,16,20,24);(2,5)(3,7)(9,11,12,20,13,18,15,23)(10,16,14,21,17,19,22,24);(1,3,4,7)(2,6,5,8)(10,17)(11,18)(14,22)(20,23);(2,5)(6,8)(10,17)(14,22)(16,19)(21,24)Polycyclic presentation:
2^9|b^2=fi,c^2=f,d^2=hi,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=di,e^b=eh,d^c=dh,g^c=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6421470);