Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417589 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
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 | |
1508 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 28 points:
(1,2,4,3)(5,6)(7,8)(10,11)(13,14)(15,19)(16,23)(17,26)(18,21)(20,24)(22,28)(25,27);(2,3)(9,10)(11,12)(13,15,18,17)(14,19,21,26)(16,24,25,22)(20,27,28,23);(9,10,12,11)(13,16,17,24,18,25,15,22)(14,20,19,23,21,28,26,27);(1,3,4,2)(6,8)(9,11,12,10)(16,25)(20,28)(22,24)(23,27)Polycyclic presentation:
2^9|a^2=egi,b^2=i,c^2=f,d^2=e,f^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,c^b=cg,d^b=de,d^c=di74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 341401);