Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32712 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C23 | C22×C22⋊C4 |
| prime radical | C24 | C22×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C2×#376 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
974 conjugacy classes of subgroups; only the 242 normal subgroups are shown.
This group has 974 conjugacy classes of subgroups — too many to draw. Only its 242 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2,4,6)(3,5,7,8)(9,10)(11,15)(12,16)(13,14)(17,23)(18,21)(19,22)(20,24)(25,26)(27,31)(28,32)(29,34)(30,33)(35,38)(36,39)(37,40);(1,3)(2,5)(4,7)(6,8)(10,14)(12,19)(15,21)(17,20)(25,27,29,36,30,28,37,35)(26,31,34,39,33,32,40,38);(2,6)(5,8)(9,11,13,18)(10,15,14,21)(12,20,19,17)(16,24,22,23)(25,28)(26,32)(27,30)(29,35)(31,33)(34,38)(36,37)(39,40);(3,7)(5,8)(9,12,13,19)(10,16,14,22)(11,17,18,20)(15,23,21,24)(26,33)(31,32)(34,40)(38,39)Polycyclic presentation:
2^9|a^2=f,b^2=h,c^2=e,d^2=e,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=di,c^b=cg,d^b=def,g^b=gi,d^c=de,g^c=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 299106);