Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #483928 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C4 |
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 | C22×#135 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
697 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2)(3,7)(4,6)(5,8)(11,19,21,36)(12,23)(13,27)(15,29,30,40)(17,32)(18,31)(20,33,34,38)(22,25,28,37)(24,39)(26,35);(1,3)(2,5)(4,8)(6,7)(9,10)(11,20,21,34)(12,24)(13,18)(14,16)(15,22,30,28)(17,26)(19,33,36,38)(23,39)(25,40,37,29)(27,31)(32,35);(3,8)(5,7)(9,11)(10,15)(12,25)(13,28)(14,21)(16,30)(17,33)(18,34)(19,35)(20,31)(22,27)(23,36)(24,38)(26,37)(29,39)(32,40);(2,6)(5,7)(10,16)(15,30)(17,24)(18,31)(20,34)(29,40)(32,39)(33,38)Polycyclic presentation:
2^9|a^2=g,b^2=g,f^2=h,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,h^a=hi,c^b=ceh,d^b=di,f^b=fi,f^c=fhi,g^c=gi,h^c=hi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72343241, #72343243, #72343246, … |
| maximal normal subgroup of | #72343241, #72343243, #72343246, … |
| maximal quotient of | #72343701, #72343712, #72343724 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 404030);