Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #153255 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2233 |
| commutator, Frattini, cosocle | C22×C16 | C23 |
| prime radical | C2×#2218 | C2 |
| LCS | G ▷ C22×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C23×C8 ◁ C22×M32 ◁ G | |
| LECS | G ▷ C22×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
294 conjugacy classes of subgroups; only the 68 normal subgroups are shown.
This group has 294 conjugacy classes of subgroups — too many to draw. Only its 68 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:
(6,11)(7,15)(8,18)(9,21)(12,24)(13,29)(14,30)(16,31)(17,35)(19,25)(20,26)(22,34)(23,27)(28,36)(32,33);(1,2)(3,4)(7,16)(11,24)(15,31)(17,26)(19,32)(25,33)(27,30)(29,36);(2,4)(5,6,9,23,10,12,21,14)(7,17,19,29,16,26,32,36)(8,20,18,13,22,35,34,28)(11,25,30,15,24,33,27,31)Polycyclic presentation:
2^9|c^2=dh,e^2=gh,g^2=hi,h^2=i,b^a=bd,c^a=ce,e^a=eg,g^a=gh,h^a=hi,c^b=cf,e^b=ei,d^c=di,e^c=egi,g^c=gh,h^c=hi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72224983, #72224998, #72225002, … |
| maximal normal subgroup of | #72224983, #72224998, #72225002, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 60451);