Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #13657 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2177 |
| commutator | C16 | C2×C42 |
| Frattini, cosocle | C22×C16 | C23 |
| prime radical | C22×D16 | C22 |
| LCS | G ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×C8 ◁ C2×#346 ◁ G | |
| LECS | G ▷ C22×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
312 conjugacy classes of subgroups; only the 106 normal subgroups are shown.
This group has 312 conjugacy classes of subgroups — too many to draw. Only its 106 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)(6,12,14,30)(7,18)(8,23)(9,26)(10,21)(13,28,17,32)(15,34,33,16)(19,25)(20,27)(22,24)(29,35,36,31);(2,4)(5,6,8,15,9,16,7,12,21,13,18,31,20,29,23,32,11,14,24,33,27,34,19,30,10,17,25,35,26,36,22,28);(6,13,14,17)(7,19)(8,24)(12,28,30,32)(15,35,33,31)(16,29,34,36)(18,25)(22,23)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=e,d^2=ghi,f^2=g,g^2=hi,h^2=i,b^a=bd,d^a=dg,f^a=fghi,g^a=gh,h^a=hi,c^b=ch,e^b=ei,d^c=di,f^c=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72216258, #72216321, #72216330, … |
| maximal normal subgroup of | #72216258, #72216321, #72216330, … |
| maximal quotient of | #72223651, #72223677, C2×#151969, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 59165);