Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #144607 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2029 |
| commutator, Frattini, cosocle | C22×C4⋊C4 | C23 |
| prime radical | #3492 | C22 |
| LCS | G ▷ C22×C4⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C22×C4⋊C4 ◁ G | |
| LECS | G ▷ C22×C4⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
509 conjugacy classes of subgroups; only the 52 normal subgroups are shown.
This group has 509 conjugacy classes of subgroups — too many to draw. Only its 52 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:
(1,2,8,23)(3,12,16,32)(4,10,24,19)(5,7,11,6)(9,25,27,18)(13,28,15,31)(14,21,30,26)(17,22,20,29);(3,9,13,21)(4,17)(10,22)(12,30,28,25)(14,31,18,32)(15,26,16,27)(19,29)(20,24);(1,3)(2,9)(4,18)(5,21)(6,13)(7,16)(8,15)(10,28)(11,27)(12,22)(14,17)(19,32)(20,25)(23,26)(24,30)(29,31)Polycyclic presentation:
2^9|a^2=dgi,b^2=d,e^2=g,f^2=g,b^a=bd,c^a=ce,f^a=fghi,c^b=cf,e^b=eh,d^c=dg,e^c=eg,f^c=fg,h^c=hi,f^e=fi44 conjugacy classes — view the full character table.
| maximal subgroup of | #72266619, #72324323, #72324331, … |
| maximal normal subgroup of | #72266619, #72324323, #72324331, … |
| maximal quotient of | #72266620, #72266623 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 51821);