Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2031 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C42 | #124 |
| commutator | C22×C4 | C22×C8 |
| Frattini, cosocle | C22×C42 | C23 |
| socle | C22 | #1441 |
| prime radical | C24 | C2×C4⋊C4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C42 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C42 ▷ C22 ▷ C1 | |
558 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 40 points:
(1,2,3,5,4,6,7,8)(9,10)(11,21)(12,25)(13,19)(14,18)(15,20)(16,23)(17,27)(22,32)(24,35)(26,34)(28,33)(29,31)(30,38)(36,37)(39,40);(9,11,13,23,15,24,14,29)(10,16,18,21,20,31,19,35)(12,22,26,37,28,38,27,39)(17,32,33,40,34,30,25,36);(9,12,13,26,15,28,14,27)(10,17,18,33,20,34,19,25)(11,22,23,37,24,38,29,39)(16,30,21,36,31,32,35,40)Polycyclic presentation:
2^9|a^2=g,b^2=d,c^2=d,d^2=h,e^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,f^a=fh,c^b=cf,e^b=eh104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 27365);