Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #404632 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2644 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C23×D8 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1050 conjugacy classes of subgroups; only the 234 normal subgroups are shown.
This group has 1050 conjugacy classes of subgroups — too many to draw. Only its 234 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6,12,8,11,16,10,15)(7,17,20,13,22,23,21,18)(9,24,27,30,26,29,25,14)(19,28,33,36,35,32,34,31);(1,2)(3,4)(5,7,12,20,11,22,10,21)(6,13,8,23,16,18,15,17)(9,19,27,33,26,35,25,34)(14,28,24,36,30,32,29,31);(2,4)(5,8,10,16,11,15,12,6)(7,18,20,23,22,13,21,17)(9,24,25,30,26,29,27,14)(19,32,33,36,35,28,34,31);(5,9,12,27,11,26,10,25)(6,14,15,29,16,30,8,24)(7,19,20,33,22,35,21,34)(13,28,17,31,18,32,23,36)Polycyclic presentation:
2^9|a^2=ef,b^2=ef,c^2=e,d^2=hi,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,h^a=hi,c^b=cg,e^b=ei,f^b=fi,f^c=fi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 320320);