Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #1343 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | #99 |
| commutator | C22×C4 | C2×C42 |
| Frattini, cosocle | C22×C22⋊C4 | C23 |
| socle | C23 | C2×#99 |
| prime radical | C23×D4 | D4 |
| LCS | G ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C23×C4 ◁ C22×C4×D4 ◁ G | |
| LECS | G ▷ C22×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
1050 conjugacy classes of subgroups; only the 162 normal subgroups are shown.
This group has 1050 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) = 20 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 20 points:
(1,2,3,4)(5,6)(7,11)(8,10)(9,12)(14,17)(15,19)(18,20);(6,10,12,11)(13,14)(15,17,19,20)(16,18);(5,7,9,8)(6,11,12,10)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=gi,d^2=g,b^a=bd,d^a=dg,f^a=fgh,d^b=dh,h^b=hi,f^d=fi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 34726);