Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #128001 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#342 |
| commutator | C24 | C2×C42 |
| Frattini, cosocle | C22×C22⋊C4 | C23 |
| prime radical | C26 | C2×C4 |
| LCS | G ▷ C24 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C25 ◁ C23×C22⋊C4 ◁ G | |
| LECS | G ▷ C22×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
1070 conjugacy classes of subgroups; only the 134 normal subgroups are shown.
This group has 1070 conjugacy classes of subgroups — too many to draw. Only its 134 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2)(3,11,13,18)(4,10)(5,9)(6,8)(7,12,19,26)(14,29,28,20)(15,22)(16,23)(17,21)(24,30)(25,32,31,27);(1,3,4,14,5,13,16,28)(2,7,10,20,9,19,23,29)(6,12,15,27,17,26,30,32)(8,18,22,31,21,11,24,25);(2,8)(7,18)(9,21)(10,22)(11,19)(20,31)(23,24)(25,29)Polycyclic presentation:
2^9|a^2=e,b^2=f,f^2=g,b^a=bd,c^a=ci,d^a=dg,f^a=fh,d^b=dh,e^b=eg,h^b=hi,f^d=fi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72235800, #72235808, #72235820, … |
| maximal normal subgroup of | #72235800, #72235808, #72235820, … |
| maximal quotient of | #72237872, #72237879, #72284242, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 35353);