Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #142540 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2021 |
| commutator, Frattini, cosocle | C22×C4⋊C4 | C23 |
| prime radical | #18164 | C2 |
| LCS | G ▷ C22×C4⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C22×C4⋊C4 ◁ G | |
| LECS | G ▷ C22×C4⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
700 conjugacy classes of subgroups; only the 58 normal subgroups are shown.
This group has 700 conjugacy classes of subgroups — too many to draw. Only its 58 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:
(2,8)(3,13)(4,17)(6,20)(9,25)(10,29)(11,18)(12,14)(21,27)(24,32)(26,31)(28,30);(1,2)(3,14)(4,18)(5,11)(6,21)(7,12)(8,17)(9,16)(10,24)(13,31)(15,26)(19,29)(20,25)(22,28)(23,27)(30,32);(1,3,15,4)(2,9,26,29)(5,13,7,17)(6,22,32,23)(8,25,31,10)(11,27,12,28)(14,30,18,21)(16,20,19,24)Polycyclic presentation:
2^9|c^2=gi,d^2=g,f^2=i,b^a=bd,c^a=ce,d^a=dg,f^a=fghi,c^b=cf,d^b=dg,e^b=eg,f^b=fi,h^b=hi,d^c=dh,f^c=fi,f^d=fi44 conjugacy classes — view the full character table.
| maximal subgroup of | #72277942, #72278278 |
| maximal normal subgroup of | #72277942, #72278278 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 49753);