Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #509 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C8 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C42 | C2×D8 |
| commutator | C8 | C2×C4×C8 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | C4×#102 |
| prime radical | C2×#135 | C2×C4 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C42 ◁ C2×C4×C8 ◁ C2×C82 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
348 conjugacy classes of subgroups; only the 142 normal subgroups are shown.
This group has 348 conjugacy classes of subgroups — too many to draw. Only its 142 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:
(1,2,4,7,5,3,6,8)(9,10)(11,20)(12,25)(13,28)(14,30)(15,19)(16,22)(17,27)(18,26)(21,34)(23,31)(24,40)(29,33)(32,39)(35,38)(36,37);(9,11,13,23,26,37,27,38,15,21,12,22,14,24,29,39)(10,16,17,20,30,35,28,40,19,31,33,34,18,32,25,36);(1,3,4,8,5,2,6,7)(11,21)(16,31)(20,34)(22,23)(24,37)(32,35)(36,40)(38,39)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=e,d^2=gi,e^2=h,f^2=gi,g^2=i,b^a=bd,d^a=dg,f^a=fgi,g^a=gi,c^b=ci128 conjugacy classes — view the full character table.
| maximal subgroup of | C8×#2179, C2×#128669 |
| maximal normal subgroup of | C8×#2179, C2×#128669 |
| maximal quotient of | C2×#128669 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 35865);