Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18284 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#120 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C22 | C2×#386 |
| prime radical | C2×#532 | C22 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1148 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2,5,8,6,3,4,7)(9,10)(11,20,23,32)(12,24)(13,19)(14,28)(15,18)(16,22,31,27)(17,25)(21,36,39,30)(26,34)(29,33)(35,37,40,38);(1,3,6,2)(4,7,5,8)(9,11,14,22,15,23,13,27)(10,16,28,20,18,31,19,32)(12,21,29,37,26,39,25,38)(17,30,24,35,33,36,34,40);(1,2,6,3)(4,8,5,7)(9,12,15,26)(10,17,18,33)(11,21,23,39)(13,25,14,29)(16,30,31,36)(19,34,28,24)(20,35,32,40)(22,37,27,38);(10,18)(16,31)(17,33)(19,28)(20,32)(24,34)(30,36)(35,40)Polycyclic presentation:
2^9|a^2=g,b^2=egh,c^2=h,e^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=dhi,f^a=fhi,e^b=eh,f^b=fi,g^b=gh,e^c=ei,f^c=fh,g^c=gi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 102589);