Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #13667 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2184 |
| commutator | C2×C16 | C22×C4 |
| Frattini, cosocle | C22×C16 | C23 |
| prime radical | C2×#508 | C22 |
| LCS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×C8 ◁ C2×#346 ◁ G | |
| LECS | G ▷ C22×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
275 conjugacy classes of subgroups; only the 82 normal subgroups are shown.
This group has 275 conjugacy classes of subgroups — too many to draw. Only its 82 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)(6,12)(7,16)(8,22)(9,19)(10,25)(13,30)(14,32)(15,29)(17,24)(18,34)(20,26)(21,36)(23,33)(27,28)(31,35);(1,3,4,2)(5,6)(7,17)(8,18)(9,24)(10,27)(11,15)(12,16)(13,19)(14,25)(20,31)(21,35)(22,29)(23,30)(26,34)(28,36)(32,33);(5,6)(7,18)(8,17)(9,24)(10,27)(11,15)(12,22)(13,19)(14,25)(16,29)(20,31)(21,30)(23,35)(26,34)(28,33)(32,36)Polycyclic presentation:
2^9|a^2=f,b^2=f,d^2=gh,e^2=gh,g^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eg,g^a=gh,h^a=hi,d^b=dg,e^b=egi,g^b=gh,h^b=hi,d^c=dgi,e^c=eg,g^c=gh,h^c=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 59637);