Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5955303 |
| automorphism | Aut(G) order 67108864 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
5042 conjugacy classes of subgroups; only the 722 normal subgroups are shown.
This group has 5042 conjugacy classes of subgroups — too many to draw. Only its 722 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:
(5,6,8,7)(10,13)(11,15)(18,21)(19,23);(2,4)(6,7)(10,13)(14,16)(18,21)(22,24);(1,2,3,4)(5,7,8,6)(17,18)(19,22)(20,21)(23,24);(9,10,12,13)(11,14,15,16)(17,18)(19,22)(20,21)(23,24);(9,11)(10,14)(12,15)(13,16)(17,19,20,23)(18,22,21,24)Polycyclic presentation:
2^9|a^2=f,c^2=gh,d^2=gi,e^2=g,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, 5863563);