Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4307937 |
| automorphism | Aut(G) order 201326592 |
| 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 | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3630 conjugacy classes of subgroups; only the 1034 normal subgroups are shown.
This group has 3630 conjugacy classes of subgroups — too many to draw. Only its 1034 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:
(2,4)(6,7)(11,15)(14,16)(17,18,20,22)(19,23,24,21);(5,6,8,7)(10,13)(14,16);(1,2,3,4)(5,7,8,6)(9,10)(11,14)(12,13)(15,16);(17,19,20,24)(18,21,22,23);(9,11)(10,14)(12,15)(13,16)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=g,d^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4215671);