Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #153090 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2184 |
| commutator, Frattini, cosocle | C22×C16 | C23 |
| LCS | G ▷ C22×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×C8 ◁ C2×#346 ◁ G | |
| LECS | G ▷ C22×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
391 conjugacy classes of subgroups; only the 76 normal subgroups are shown.
This group has 391 conjugacy classes of subgroups — too many to draw. Only its 76 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) = 128 |
A minimal faithful permutation representation — 3 generators on 36 points:
(6,14)(7,21)(8,25)(9,30)(10,12)(11,15)(16,19)(17,32)(18,22)(20,26)(23,31)(24,36)(27,33)(28,34)(29,35);(1,2)(3,4)(5,6)(7,11)(8,21)(9,31)(10,19)(12,17)(13,20)(14,25)(15,18)(16,34)(22,28)(23,33)(24,27)(26,35)(29,36)(30,32);(2,4)(5,7)(6,15)(8,26)(9,32)(10,23)(11,19)(12,18)(13,24)(14,29)(16,28)(17,27)(20,33)(21,30)(22,35)(25,31)(34,36)Polycyclic presentation:
2^9|d^2=gh,e^2=ghi,f^2=i,g^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=egi,f^a=fi,g^a=gh,h^a=hi,c^b=cf,d^b=dg,e^b=eg,f^b=fi,g^b=gh,h^b=hi,d^c=dgi,e^c=egi,f^c=fi,g^c=gh,h^c=hi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 60305);