Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #153213 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2226 |
| commutator, Frattini, cosocle | C22×C16 | C23 |
| prime radical | C2×#508 | C22 |
| 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 | |
263 conjugacy classes of subgroups; only the 72 normal subgroups are shown.
This group has 263 conjugacy classes of subgroups — too many to draw. Only its 72 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:
(6,12,14,28)(7,17)(8,10)(9,23)(13,31,16,26)(15,33,32,36)(18,24,30,29)(19,25)(20,35)(21,34)(22,27);(1,2)(3,4)(6,13,14,16)(12,26,28,31)(15,29,32,24)(18,33,30,36);(2,4)(5,6)(7,18)(8,16)(9,24)(10,13)(11,14)(12,27)(15,23)(17,28)(19,32)(20,36)(21,33)(22,30)(25,29)(26,34)(31,35)Polycyclic presentation:
2^9|a^2=d,b^2=d,e^2=ghi,f^2=i,g^2=hi,h^2=i,b^a=bd,c^a=ce,e^a=eg,f^a=fi,g^a=gh,h^a=hi,c^b=cf,d^c=di,e^c=egi,f^c=fi,g^c=gh,h^c=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72224983, #72225008, #72225016 |
| maximal normal subgroup of | #72224983, #72225008, #72225016 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 60409);