Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #142228 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2021 |
| commutator, Frattini, cosocle | C2×#117 | C23 |
| prime radical | #18300 | C2 |
| LCS | G ▷ C2×#117 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C2×#117 ▷ C23 ▷ C2 ▷ C1 | |
613 conjugacy classes of subgroups; only the 70 normal subgroups are shown.
This group has 613 conjugacy classes of subgroups — too many to draw. Only its 70 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)(4,6)(5,8)(9,10)(11,20)(12,17)(13,16)(14,19)(15,22)(18,28)(21,37)(23,36)(24,35)(25,32)(26,33)(27,34)(29,39)(30,38)(31,40);(2,6)(3,4)(7,8)(9,11)(10,15)(12,22)(13,21)(14,24)(16,30)(17,20)(18,29)(19,31)(23,28)(25,38)(26,39)(27,40)(32,37)(33,36)(34,35);(1,3,5,4)(2,7,8,6)(10,16)(11,21)(13,28)(17,32)(18,19)(20,35)(22,39)(23,24)(25,26)(29,30)(33,34)(38,40)Polycyclic presentation:
2^9|c^2=g,d^2=g,e^2=i,f^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,f^a=fhi,c^b=cf,d^b=dg,f^b=fi,h^b=hi,d^c=dh,e^c=ei,f^c=fi,f^d=fi44 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 49472);