Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23391 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
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×C4⋊C4 |
| prime radical | C24 | C22×Q8 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C2×C43 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1038 conjugacy classes of subgroups; only the 386 normal subgroups are shown.
This group has 1038 conjugacy classes of subgroups — too many to draw. Only its 386 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) = 256 |
A minimal faithful permutation representation — 4 generators on 28 points:
(2,4)(13,14,17,20,18,21,16,26)(15,22,25,27,23,28,24,19);(5,6)(7,10)(8,9)(11,12)(13,15,16,24,18,23,17,25)(14,19,20,22,21,27,26,28);(1,2,3,4)(6,9)(10,12);(5,7,8,11)(6,10,9,12)(15,23)(19,27)(22,28)(24,25)Polycyclic presentation:
2^9|a^2=h,b^2=e,c^2=f,d^2=g,e^2=i,h^2=i,b^a=be,c^a=cf,e^a=ei,c^b=cg,d^b=di104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 266180);