Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #153105 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2206 |
| commutator, Frattini, cosocle | C22×C16 | C23 |
| LCS | G ▷ C22×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×C8 ◁ C2×C4×C16 ◁ G | |
| LECS | G ▷ C22×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
419 conjugacy classes of subgroups; only the 76 normal subgroups are shown.
This group has 419 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) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(1,2)(3,4)(5,6)(7,19)(8,13)(9,16)(10,27)(11,28)(12,18)(14,21)(15,23)(17,22)(20,32)(24,31)(25,29)(26,34)(30,36)(33,35);(2,4)(6,13)(7,20)(8,23)(9,11)(10,22)(14,32)(15,17)(16,34)(18,31)(19,33)(21,26)(24,35)(25,36)(27,30)(28,29);(5,7)(6,14)(8,24)(9,10)(11,20)(12,22)(13,19)(15,16)(17,31)(18,30)(21,23)(27,29)(28,33)(32,34)(35,36)Polycyclic presentation:
2^9|d^2=gh,e^2=gh,f^2=i,g^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eg,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=dg,e^c=eg,f^c=fi,g^c=gh,h^c=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72224983, #72224996, #72225000 |
| maximal normal subgroup of | #72224983, #72224996, #72225000 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 60301);