Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #124940 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1519 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C22×M16 | C23 |
| prime radical | C2×#469 | C22 |
| LCS | G ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C22×M16 ▷ C2×C4 ▷ C2 ▷ C1 | |
444 conjugacy classes of subgroups; only the 94 normal subgroups are shown.
This group has 444 conjugacy classes of subgroups — too many to draw. Only its 94 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(2,4)(5,6)(7,11,17,26)(8,16)(9,22,20,12)(10,14)(13,23,29,34)(15,21)(18,32)(19,27,31,28)(24,30)(25,36)(33,35);(1,2,3,4)(5,7)(6,11)(8,17)(9,18)(10,19)(12,24)(13,25)(14,27)(15,28)(16,26)(20,30)(21,31)(22,32)(23,33)(29,35)(34,36);(6,12)(9,23)(10,21)(11,24)(13,15)(14,29)(16,22)(18,33)(19,31)(20,34)(25,28)(26,32)(27,35)(30,36)Polycyclic presentation:
2^9|a^2=f,b^2=dgh,e^2=hi,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=eg,h^a=hi,f^b=fi,e^c=eh,f^c=fghi,g^c=gi,h^c=hi,f^e=fi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72265345, #72265422, #72307220, … |
| maximal normal subgroup of | #72265345, #72265422, #72307220, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 41718);