Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4452694 |
| automorphism | Aut(G) order 8388608 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C22×#392 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
5466 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)(5,6,8,7)(9,10,13,11)(12,15,16,14)(17,18,21,24)(19,23,20,22);(6,7)(9,10)(11,13)(12,14)(15,16)(17,19,21,20)(18,22,24,23);(6,7)(10,11)(14,15)(19,20)(22,23);(1,2)(3,4)(5,7,8,6)(9,11,13,10)(12,15,16,14)(17,20,21,19)(18,23,24,22);(9,12)(10,14)(11,15)(13,16)(17,18)(19,22)(20,23)(21,24)Polycyclic presentation:
2^9|a^2=f,b^2=fg,d^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dg,d^c=df80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4359909);