Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #13657 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2177 |
| commutator | C16 | C22×C8 |
| Frattini, cosocle | C4×C16 | C23 |
| prime radical | C2×D16 | C2×C4 |
| LCS | G ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ C82 ◁ #1311 ◁ G | |
| LECS | G ▷ C4×C16 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
146 conjugacy classes of subgroups; only the 60 normal subgroups are shown.
This group has 146 conjugacy classes of subgroups — too many to draw. Only its 60 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)(9,10,12,19,14,21,15,17)(11,22,26,39,24,16,28,32)(13,29,25,37,31,35,30,20)(18,36,33,23,38,40,34,27);(10,16)(11,23)(12,15)(13,30)(17,32)(18,37)(19,39)(20,38)(21,22)(24,27)(25,31)(26,36)(28,40)(29,33)(34,35);(1,3,4,8,5,2,6,7)(10,17,21,19)(11,24)(16,32,22,39)(18,33,38,34)(20,35,37,29)(23,27)(26,28)(36,40)Polycyclic presentation:
2^9|a^2=e,c^2=eh,d^2=fhi,e^2=g,f^2=hi,h^2=i,b^a=bd,c^a=ch,d^a=df,f^a=fh,h^a=hi,d^b=dfi,e^b=ei,f^b=fh,h^b=hi,d^c=di68 conjugacy classes — view the full character table.
| maximal subgroup of | #72223673, C2×#151083, #72232000, … |
| maximal normal subgroup of | #72223673, C2×#151083, #72232000, … |
| maximal quotient of | C2×#151083, #72286481, #72286684 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 58279);