Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10326 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #392 |
| commutator | C23×C4 | C22×C4 |
| Frattini, cosocle | C24×C4 | C23 |
| prime radical | C25 | C2×D4 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C22 ▷ C1 | |
1308 conjugacy classes of subgroups; only the 162 normal subgroups are shown.
This group has 1308 conjugacy classes of subgroups — too many to draw. Only its 162 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 28 points:
(1,2)(3,4)(5,6)(7,11,12,9)(8,10)(14,19,21,22)(20,27,28,23);(5,7,8,12)(6,9,10,11)(13,14)(15,23)(16,22)(17,19)(18,21)(20,25)(24,28)(26,27);(2,4)(5,6)(7,9)(8,10)(11,12)(13,15,18,26)(14,20,21,28)(16,25,17,24)(19,23,22,27)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=i,d^2=i,f^2=i,b^a=bd,c^a=ce,d^a=dh,c^b=cf,d^b=di,e^b=ehi,f^b=fi,g^b=ghi,d^c=dhi,f^c=fi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 18650);