Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #150243 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#357 |
| commutator | C2×C16 | C22×C4 |
| Frattini, cosocle | C4×C16 | C23 |
| prime radical | C23 | #357 |
| LCS | G ▷ C2×C16 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C2×C4×C8 ◁ C2×C4×C16 ◁ G | |
| LECS | G ▷ C4×C16 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
202 conjugacy classes of subgroups; only the 92 normal subgroups are shown.
This group has 202 conjugacy classes of subgroups — too many to draw. Only its 92 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) = 256 |
A minimal faithful permutation representation — 3 generators on 40 points:
(2,5)(3,7)(6,8)(9,10,14,20,32,23,38,39,17,19,27,34,18,25,33,36,15,21,13,30,29,35,37,40,12,26,28,31,16,11,22,24);(1,2,4,6)(3,8,7,5)(9,11,15,25)(10,18,21,16)(12,23,17,35)(13,31,14,34)(19,32,26,29)(20,28,30,27)(22,36,33,24)(37,40,38,39);(9,12,15,17)(10,19,21,26)(11,23,25,35)(13,27,14,28)(16,29,18,32)(20,34,30,31)(22,37,33,38)(24,39,36,40)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=i,d^2=gh,e^2=fghi,g^2=hi,h^2=i,b^a=bd,c^a=ci,d^a=df,d^b=dg,e^b=efg,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, 58156);