Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #465 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C8 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4×C8 | C2×D4 |
| commutator | C2×C4 | C2×C4×C8 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | C8×C4⋊C4 |
| prime radical | C23 | #405 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C8 ◁ C2×C4×C16 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
300 conjugacy classes of subgroups; only the 166 normal subgroups are shown.
This group has 300 conjugacy classes of subgroups — too many to draw. Only its 166 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:
(9,10,14,19,15,20,32,29,16,21,33,28,13,18,27,12)(11,22,25,23,24,36,39,38,26,37,31,40,30,17,34,35);(1,2,3,6,4,5,7,8)(9,11)(10,17)(12,23)(13,30)(14,31)(15,24)(16,26)(18,37)(19,38)(20,22)(21,36)(25,33)(27,39)(28,35)(29,40)(32,34);(2,5)(6,8)(9,12,27,18,13,28,33,21,16,29,32,20,15,19,14,10)(11,23,39,37,30,35,25,36,26,40,34,22,24,38,31,17)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=de,d^2=i,e^2=g,f^2=h,g^2=i,b^a=bd,c^b=ch,d^b=di,e^b=ei176 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 36255);