Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #454382 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C23×D4 | D4 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C22×C4×D4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1606 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 28 points:
(5,6)(7,8)(10,11)(13,14,19,23)(15,20,18,27)(16,25,26,22)(17,21,24,28);(1,2,3,4)(9,10,12,11)(13,15,19,18)(14,20,23,27)(16,17,26,24)(21,22,28,25);(2,4)(9,10)(11,12)(13,16,15,24,19,26,18,17)(14,21,27,22,23,28,20,25);(6,8)(9,11,12,10)(13,17,18,26,19,24,15,16)(14,22,20,21,23,25,27,28)Polycyclic presentation:
2^9|a^2=i,b^2=g,c^2=efi,d^2=f,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,h^a=hi,c^b=cg,d^b=di,d^c=de68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 373945);