Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #150785 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2×C4 |
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 | #33713 | C2 |
| 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 | |
279 conjugacy classes of subgroups; only the 78 normal subgroups are shown.
This group has 279 conjugacy classes of subgroups — too many to draw. Only its 78 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:
(1,2,3,6,4,5,7,8)(9,10,13,19,30,22,38,40,16,17,32,29,33,25,37,23,14,20,31,18,28,34,36,39,12,26,27,35,15,11,21,24);(2,5)(3,7)(9,11)(10,15)(12,22)(13,29)(14,25)(16,34)(17,28)(18,27)(19,32)(20,33)(21,23)(24,37)(26,30)(31,35)(36,39)(38,40);(2,5)(6,8)(9,12,14,16)(10,17,20,26)(11,22,25,34)(13,27,31,32)(15,28,33,30)(18,35,19,29)(21,36,37,38)(23,39,24,40)Polycyclic presentation:
2^9|a^2=e,c^2=i,d^2=gh,e^2=fghi,g^2=hi,h^2=i,b^a=bd,c^a=cf,d^a=df,d^b=dg,e^b=efg,g^b=gh,h^b=hi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 58183);