Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2021 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C42 | #120 |
| commutator | C24 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | C2×#379 |
| prime radical | #62686 | C2 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C42 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2540 conjugacy classes of subgroups; only the 374 normal subgroups are shown.
This group has 2540 conjugacy classes of subgroups — too many to draw. Only its 374 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 — 4 generators on 40 points:
(9,10,13,18,14,17,26,31)(11,19,22,32,21,15,25,28)(12,16,24,29,23,30,36,40)(20,27,35,38,34,33,37,39);(1,2)(3,6)(4,5)(7,8)(9,11)(10,15)(12,20)(13,25)(14,21)(16,27)(17,19)(18,32)(22,26)(23,34)(24,37)(28,31)(29,39)(30,33)(35,36)(38,40);(2,5)(6,8)(9,12)(10,16)(11,20)(13,24)(14,23)(15,27)(17,30)(18,29)(19,33)(21,34)(22,35)(25,37)(26,36)(28,38)(31,40)(32,39);(1,3,4,7)(2,6,5,8)Polycyclic presentation:
2^9|a^2=h,d^2=g,h^2=i,b^a=be,c^a=cf,e^a=ei,c^b=cg,f^b=fi,h^b=hi,e^c=ei104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 266061);