Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #124940 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
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 | |
436 conjugacy classes of subgroups; only the 94 normal subgroups are shown.
This group has 436 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)(5,6,9,16,11,14,10,25)(7,17,19,29,21,12,20,28)(8,13,22,15,24,30,23,31)(18,32,35,27,36,26,34,33);(1,2,3,4)(5,7)(6,12)(8,18)(9,20)(10,19)(11,21)(13,26)(14,17)(15,27)(16,29)(22,34)(23,35)(24,36)(25,28)(30,32)(31,33);(6,13)(8,22)(9,10)(12,26)(14,30)(15,16)(17,32)(18,34)(19,20)(23,24)(25,31)(27,29)(28,33)(35,36)Polycyclic presentation:
2^9|a^2=f,b^2=dgh,e^2=hi,f^2=i,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 | #72265006, #72265060, #72307213, … |
| maximal normal subgroup of | #72265006, #72265060, #72307213, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 41720);