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 | |
3219 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 24 points:
(9,10,14,17,15,18,13,16)(11,19,20,23,22,24,21,12);(1,2)(3,7)(4,6)(5,8)(9,11)(10,12)(13,20)(14,21)(15,22)(16,19)(17,24)(18,23);(1,3)(2,5)(4,8)(6,7)(9,10)(11,12)(13,16)(14,17)(15,18)(19,20)(21,24)(22,23);(3,8)(5,7)(9,12)(10,11)(13,24)(14,19)(15,23)(16,21)(17,20)(18,22);(2,6)(5,7)(9,13,15,14)(10,16,18,17)(11,20,22,21)(12,19,23,24)Polycyclic presentation:
2^9|a^2=f,e^2=i,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, 6407428);