Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #485617 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×B(4,𝔽2) |
| commutator, Frattini, cosocle | C2×C22⋊C4 | C24 |
| prime radical | #45458 | C2 |
| LCS | G ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C22×C22⋊C4 ◁ G | |
| LECS | G ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
1292 conjugacy classes of subgroups; only the 122 normal subgroups are shown.
This group has 1292 conjugacy classes of subgroups — too many to draw. Only its 122 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 — 4 generators on 32 points:
(1,2,5,9)(3,11,13,22)(4,15,18,8)(6,10,19,24)(7,16,21,14)(12,25,26,32)(17,28,30,23)(20,29,31,27);(1,3)(2,7)(4,16)(5,13)(6,12)(8,22)(9,21)(10,20)(11,15)(14,18)(17,29)(19,26)(23,32)(24,31)(25,28)(27,30);(3,12)(7,20)(8,23)(13,26)(14,27)(15,28)(16,29)(21,31);(3,13)(7,21)(11,22)(12,26)(14,16)(20,31)(25,32)(27,29)Polycyclic presentation:
2^9|a^2=g,e^2=g,b^a=be,c^a=cf,e^a=eg,h^a=hi,c^b=ci,d^b=dg,e^b=eg,f^b=fhi,e^c=eh,f^e=fi44 conjugacy classes — view the full character table.
| maximal subgroup of | #72227597, #72227715, #72284152, … |
| maximal normal subgroup of | #72227597, #72284152, #72284337, … |
| maximal quotient of | #72227592 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 392893);