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 | #33517 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
766 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2)(3,7)(4,6)(5,8)(9,10,13,19,15,20,14,30)(11,16,21,33,22,32,24,39)(12,25,27,36,29,17,28,18)(23,34,37,35,26,40,38,31);(1,3)(2,5)(4,8)(6,7)(9,11,13,21,15,22,14,24)(10,16,19,33,20,32,30,39)(12,26,27,38,29,23,28,37)(17,34,18,35,25,40,36,31);(3,8)(5,7)(10,17)(11,21)(12,27)(13,14)(16,31)(18,30)(19,36)(20,25)(22,24)(23,26)(28,29)(32,35)(33,40)(34,39);(2,6)(5,7)(11,22)(16,32)(21,24)(23,26)(31,35)(33,39)(34,40)(37,38)Polycyclic presentation:
2^9|a^2=g,b^2=g,f^2=h,g^2=i,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 | #72265445, #72265451, #72265458, … |
| maximal normal subgroup of | #72265445, #72265451, #72265458, … |
| maximal quotient of | #72320494, C2×#496839 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 404035);