Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5955303 |
| automorphism | Aut(G) order 16777216 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C26 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
4486 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(5,6,8,7)(10,13)(11,15)(17,18)(19,22)(20,24)(21,23);(2,4)(6,7)(10,13)(14,16)(17,19,21,24)(18,22,23,20);(1,2)(3,4)(5,7,8,6)(17,19)(18,20)(21,24)(22,23);(9,10)(11,14)(12,13)(15,16)(17,19)(18,20)(21,24)(22,23);(9,11)(10,14)(12,15)(13,16)(17,20)(18,19)(21,22)(23,24)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=f,b^a=bf,c^a=cg,d^a=di,e^a=ei,c^b=ch,d^b=di104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5862519);