Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #6342371 |
| automorphism | Aut(G) order 402653184 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C24×C44 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C22×#127 | C22 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
4110 conjugacy classes of subgroups; only the 978 normal subgroups are shown.
This group has 4110 conjugacy classes of subgroups — too many to draw. Only its 978 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,3)(6,7)(11,12)(13,15)(19,23)(22,24);(1,2,4,3)(5,6,8,7)(9,10,14,16)(11,15,12,13)(17,18,21,20)(19,22,23,24);(5,6,8,7)(9,11,14,12)(10,15,16,13)(17,19)(18,22)(20,24)(21,23);(5,6,8,7)(9,12,14,11)(10,13,16,15)(17,20,21,18)(19,24,23,22);(1,3,4,2)(5,7,8,6)(9,13)(10,11)(12,16)(14,15)(17,19)(18,22)(20,24)(21,23)Polycyclic presentation:
2^9|b^2=i,c^2=h,d^2=g,e^2=f,b^a=bf,c^a=cg,d^a=dh,e^a=ei152 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6249599);