Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #144613 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2029 |
| 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 | |
565 conjugacy classes of subgroups; only the 52 normal subgroups are shown.
This group has 565 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)(3,13,17,25)(4,18,22,29)(5,20,19,10)(6,11,7,12)(9,14,27,32)(15,31,24,28)(16,26,21,30);(3,14,15,21)(9,25,26,28)(13,30,31,27)(16,17,32,24);(1,3)(2,9)(4,14)(5,21)(6,15)(7,17)(8,24)(10,28)(11,26)(12,27)(13,20)(16,22)(18,31)(19,32)(23,30)(25,29)Polycyclic presentation:
2^9|a^2=dgi,b^2=d,e^2=gi,f^2=g,b^a=bd,c^a=ce,e^a=ei,f^a=fghi,c^b=cf,e^b=eh,d^c=dg,e^c=egi,f^c=fg,h^c=hi,f^e=fi44 conjugacy classes — view the full character table.
| maximal subgroup of | #72233403, #72233489, #72233490, … |
| maximal normal subgroup of | #72233403, #72233489, #72233490, … |
| maximal quotient of | #72339054, #72339058 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 51814);