Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6500226 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#120 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3340 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 24 points:
(9,10,12,15,13,17,20,16)(11,14,19,22,18,21,24,23);(1,2)(3,7)(4,6)(5,8)(9,10)(11,14)(12,16)(13,17)(15,20)(18,21)(19,23)(22,24);(1,3)(2,5)(4,8)(6,7)(9,10)(11,14)(12,15)(13,17)(16,20)(18,21)(19,22)(23,24);(3,8)(5,7)(9,11)(10,14)(12,19)(13,18)(15,22)(16,23)(17,21)(20,24);(2,6)(5,7)(11,18)(14,21)(19,24)(22,23)Polycyclic presentation:
2^9|a^2=f,f^2=i,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,e^b=eh,f^b=fi,d^c=dh,g^c=gi,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6407422);