Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #45438 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, cosocle | C24 | C25 |
| commutator | C23 | C24×C4 |
| prime radical | C22×#127 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3510 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 24 points:
(9,10)(11,13)(12,14)(15,16)(18,19)(20,24);(2,4)(5,6,8,7)(10,11)(12,16)(17,18,21,19)(20,23,24,22);(1,2)(3,4)(5,7,8,6)(17,18,21,19)(20,22,24,23);(9,11,13,10)(12,15,16,14)(17,19,21,18)(20,23,24,22);(9,12)(10,14)(11,15)(13,16)(17,20,21,24)(18,22,19,23)Polycyclic presentation:
2^9|b^2=i,c^2=i,d^2=f,e^2=g,b^a=bf,c^a=cg,d^a=df,e^a=eg,c^b=ch,d^b=dfg,e^b=ef104 conjugacy classes — view the full character table.
| Fitting subgroup of | #72258829 |
| maximal subgroup of | #72258789, #72258829, #72258897, … |
| maximal normal subgroup of | #72258789, #72258829, #72258897, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 467328);