Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #142536 |
| 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×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 | |
743 conjugacy classes of subgroups; only the 58 normal subgroups are shown.
This group has 743 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:
(1,2,8,14)(3,15,20,29)(4,21,24,23)(5,11,25,12)(6,13,9,7)(10,26,28,17)(16,27,31,18)(19,32,30,22);(1,3)(2,10)(4,22)(5,26)(6,27)(7,19)(8,20)(9,18)(11,15)(12,29)(13,30)(14,28)(16,23)(17,25)(21,31)(24,32);(1,4,6,5)(2,11,13,21)(3,16,27,29)(7,23,14,12)(8,24,9,25)(10,17,30,32)(15,20,31,18)(19,22,28,26)Polycyclic presentation:
2^9|a^2=g,c^2=gi,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 | #72231369, #72231832, #72234698, … |
| maximal normal subgroup of | #72231369, #72231832, #72234698, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 49781);