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)(9,10,13,18,28,21,36,38,15,17,29,33,31,24,37,39,16,19,30,34,26,32,35,40,12,23,25,27,14,11,20,22);(1,2,4,6)(3,8,7,5)(9,11)(10,14)(12,21)(13,27)(15,32)(16,24)(17,26)(18,25)(19,31)(20,22)(23,28)(29,34)(30,33)(35,38)(36,40)(37,39);(9,12,16,15)(10,17,19,23)(11,21,24,32)(13,25,30,29)(14,26,31,28)(18,33,34,27)(20,35,37,36)(22,38,39,40)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=i,d^2=gh,e^2=fgh,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, 58152);