Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #124648 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #418 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | #1490 |
| prime radical | #3054 | C4 |
| LCS | G ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C8 ◁ C2×C4×M16 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C22×C4 ▷ C2 ▷ C1 | |
378 conjugacy classes of subgroups; only the 72 normal subgroups are shown.
This group has 378 conjugacy classes of subgroups — too many to draw. Only its 72 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)(9,10,14,19,15,21,12,18,16,22,29,37,32,20,13,30)(11,23,25,17,27,34,24,35,28,33,39,36,26,40,31,38);(1,3)(2,5)(4,8)(6,7)(9,11)(10,17)(12,24)(13,31)(14,25)(15,27)(16,28)(18,33)(19,34)(20,38)(21,35)(22,36)(23,30)(26,32)(29,39)(37,40);(2,6)(5,7)(9,12,15,29,16,13,32,14)(10,18,21,37,22,30,20,19)(11,24,27,39,28,31,26,25)(17,33,35,40,36,23,38,34)Polycyclic presentation:
2^9|a^2=e,c^2=h,d^2=gi,e^2=h,f^2=i,g^2=i,h^2=i,b^a=bd,c^a=cfh,d^a=df,f^a=fi,g^a=gi,d^b=dg,e^b=efgi,f^b=fi,g^b=gi,d^c=di,e^c=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 32086);