Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #8453 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #379 |
| commutator | C22×C4 | C2×C42 |
| Frattini, cosocle | C24×C4 | C23 |
| prime radical | C24×D4 | C22 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C22 ▷ C1 | |
2882 conjugacy classes of subgroups; only the 226 normal subgroups are shown.
This group has 2882 conjugacy classes of subgroups — too many to draw. Only its 226 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 20 points:
(5,6)(7,11,12,9)(8,10)(14,17,18,19);(1,2,3,4)(5,7)(6,9)(8,12)(10,11)(13,14)(15,19)(16,18)(17,20);(15,20)(17,19)Polycyclic presentation:
2^9|a^2=f,b^2=g,d^2=i,b^a=bd,c^a=ce,d^a=dh,d^b=di,e^b=ei,f^b=fhi,d^c=di80 conjugacy classes — view the full character table.
| maximal subgroup of | #72219159, #72219162, #72219163, … |
| maximal normal subgroup of | #72219159, #72219162, #72219163, … |
| maximal quotient of | #72225897, #72242823, #72242855, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 14326);