Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #142536 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2021 |
| commutator, Frattini, cosocle | C22×C22⋊C4 | C23 |
| prime radical | #3490 | C22 |
| LCS | G ▷ C22×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C22×C22⋊C4 ◁ G | |
| LECS | G ▷ C22×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
689 conjugacy classes of subgroups; only the 58 normal subgroups are shown.
This group has 689 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:
(1,2,6,11)(3,13,23,32)(4,17,22,28)(5,10,7,12)(8,24,29,14)(9,21,19,20)(15,27,16,30)(18,25,31,26);(1,3)(2,8)(4,18)(5,15)(6,23)(7,16)(9,27)(10,25)(11,29)(12,26)(13,17)(14,20)(19,30)(21,24)(22,31)(28,32);(1,2,7,12)(3,8,16,26)(4,19,21,17)(5,10,6,11)(9,20,28,22)(13,24,30,18)(14,27,31,32)(15,25,23,29)Polycyclic presentation:
2^9|a^2=gi,c^2=i,d^2=g,b^a=bd,c^a=ce,d^a=dg,f^a=fghi,c^b=cf,d^b=dg,e^b=eg,h^b=hi,d^c=dh,f^d=fi44 conjugacy classes — view the full character table.
| maximal subgroup of | #72251835, #72252024, #72252027 |
| maximal normal subgroup of | #72251835, #72252024, #72252027 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 49767);