Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #13691 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2206 |
| commutator | C2×C16 | C22×C4 |
| Frattini, cosocle | C22×C16 | C23 |
| prime radical | C23 | #509 |
| LCS | G ▷ C2×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 | |
198 conjugacy classes of subgroups; only the 94 normal subgroups are shown.
This group has 198 conjugacy classes of subgroups — too many to draw. Only its 94 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:
(1,2,4,3)(6,13)(7,10)(8,20)(9,11)(14,29)(15,31)(16,28)(17,22)(18,19)(21,25)(23,33)(24,32)(26,27)(30,36)(34,35);(1,3,4,2)(5,6)(7,14)(8,21)(9,15)(10,25)(11,26)(12,16)(13,20)(17,31)(18,34)(19,30)(22,29)(23,36)(24,35)(27,33)(28,32);(5,7,11,18,12,19,9,10)(6,14,26,34,16,30,15,25)(8,17,23,32,24,33,22,20)(13,21,31,36,28,35,27,29)Polycyclic presentation:
2^9|a^2=f,b^2=f,c^2=ei,d^2=gh,e^2=i,g^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,g^a=gh,h^a=hi,d^b=dg,g^b=gh,h^b=hi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 60063);