Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #483919 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#442 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| prime radical | C2×#135 | C23 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C4×#97 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
471 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 40 points:
(2,5)(3,7)(9,10)(11,19,21,36)(12,15)(13,26)(14,29)(16,22,30,38)(17,28)(18,25)(20,37,23,33)(24,35)(27,34,39,32)(31,40);(1,2)(3,6)(4,5)(7,8)(10,15)(11,20,21,23)(13,24)(16,32,30,34)(18,29)(19,33,36,37)(22,39,38,27)(28,40);(2,5)(6,8)(9,11)(10,16)(12,21)(13,27)(14,20)(15,30)(17,33)(18,34)(19,35)(22,40)(23,25)(24,39)(26,36)(28,38)(29,32)(31,37);(1,3,4,7)(2,6,5,8)(10,15)(16,30)(17,31)(18,29)(19,36)(26,35)(32,34)(33,37)Polycyclic presentation:
2^9|a^2=g,b^2=g,d^2=ei,f^2=h,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,h^a=hi,c^b=ceh,f^b=fi,f^c=fhi,g^c=gi,h^c=hi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72313363, #72313392, #72313439, … |
| maximal normal subgroup of | #72313363, #72313392, #72313439, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 404123);