Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1812 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C8 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | #117 |
| commutator | C2×C4 | C2×C4×C8 |
| Frattini, cosocle | C23×C8 | C23 |
| socle | C23 | #432 |
| prime radical | C24 | C2×M16 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C22×#97 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
622 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)(3,4)(9,10,15,18,13,21,31,37,16,20,30,36,14,19,12,28)(11,22,26,39,24,17,23,33,27,35,40,38,25,34,29,32);(5,6,7,8)(9,11)(10,17,20,34)(12,29)(13,24)(14,25)(15,26)(16,27)(18,33,36,32)(19,22,21,35)(23,31)(28,39,37,38)(30,40);(2,4)(9,12,13,15,16,31,14,30)(10,18,19,28,20,36,21,37)(11,23,24,40,27,29,25,26)(17,32,22,38,34,33,35,39)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=d,d^2=i,e^2=i,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=di,g^a=gi,c^b=ci,f^c=fi128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 54015);