Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #150243 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#357 |
| commutator | C2×C16 | C22×C4 |
| Frattini, cosocle | C22×C16 | C23 |
| prime radical | #33600 | C2 |
| LCS | G ▷ C2×C16 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C23×C8 ◁ C23×C16 ◁ G | |
| LECS | G ▷ C22×C16 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
337 conjugacy classes of subgroups; only the 86 normal subgroups are shown.
This group has 337 conjugacy classes of subgroups — too many to draw. Only its 86 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 40 points:
(1,2)(3,7,8,5)(4,6)(10,16,17,32)(11,21)(12,27)(13,23)(14,29)(18,35,19,36)(20,37,31,34)(22,28,33,30)(24,26)(25,40)(38,39);(1,3)(2,5)(4,8)(6,7)(9,10)(11,22)(12,28)(13,30)(14,31)(15,17)(16,24)(18,27)(19,23)(20,29)(21,32)(25,35)(26,33)(34,39)(36,38)(37,40);(10,17)(16,32)(18,19)(20,31)(22,33)(28,30)(34,37)(35,36)Polycyclic presentation:
2^9|a^2=e,d^2=gh,f^2=hi,g^2=hi,h^2=i,b^a=bd,d^a=df,f^a=fh,g^a=gh,h^a=hi,c^b=ci,d^b=dg,e^b=efgh,f^b=fh,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, 57439);