Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11692 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#99 |
| commutator | C22×C4 | C22×C8 |
| Frattini, cosocle | C2×#97 | C23 |
| socle | C22 | C2×#323 |
| prime radical | C2×#121 | C8 |
| LCS | G ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ C2×C4×M16 ◁ G | |
| LECS | G ▷ C2×#97 ▷ C22×C4 ▷ C2 ▷ C1 | |
404 conjugacy classes of subgroups; only the 90 normal subgroups are shown.
This group has 404 conjugacy classes of subgroups — too many to draw. Only its 90 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) = 128 |
A minimal faithful permutation representation — 3 generators on 40 points:
(2,6,7,8)(9,10,12,17,13,18,24,32,14,16,23,29,25,30,36,39)(11,19,21,31,22,33,35,40,20,15,26,27,34,28,37,38);(1,2)(3,6)(4,8)(5,7)(9,11)(10,15)(12,21)(13,22)(14,20)(16,19)(17,31)(18,28)(23,26)(24,35)(25,34)(27,29)(30,33)(32,40)(36,37)(38,39);(1,3,5,4)(2,6,7,8)(11,20)(15,19)(21,26)(22,34)(27,31)(28,33)(35,37)(38,40)Polycyclic presentation:
2^9|a^2=e,c^2=g,d^2=g,e^2=h,h^2=i,b^a=bd,d^a=df,f^a=fi,c^b=ci,d^b=dg,e^b=efg,e^d=ei80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 31342);