Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #29742 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C43 |
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 | #36644 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C2×C43 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1770 conjugacy classes of subgroups; only the 306 normal subgroups are shown.
This group has 1770 conjugacy classes of subgroups — too many to draw. Only its 306 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 — 4 generators on 28 points:
(5,6)(7,10)(8,9)(11,12)(13,14)(15,21,23,28)(16,20)(17,19)(18,22,26,24)(25,27);(1,2,3,4)(6,9)(10,12);(2,4)(7,11)(10,12)(13,15)(14,18)(16,24)(17,23)(19,26)(20,28)(21,27)(22,25);(5,7,8,11)(6,10,9,12)(14,19)(18,26)(20,27)(21,28)Polycyclic presentation:
2^9|a^2=h,b^2=g,d^2=e,f^2=i,b^a=be,c^a=cf,d^a=di,c^b=cg,d^c=de,f^c=fi,h^c=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 287798);