Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #394289 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#121 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C25 | #41 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1226 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 28 points:
(1,2,3,4)(13,14,19,22)(15,18,20,16)(17,25,27,23)(21,26,28,24);(5,6)(7,10)(8,9)(11,12)(13,15,18,22,19,20,16,14)(17,23,26,28,27,25,24,21);(2,4)(6,9)(10,12)(13,16,19,18)(14,20,22,15)(17,24,27,26)(21,25,28,23);(5,7,8,11)(6,10,9,12)(13,17)(14,21)(15,23)(16,24)(18,26)(19,27)(20,25)(22,28)Polycyclic presentation:
2^9|a^2=f,b^2=e,c^2=i,d^2=g,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,h^d=hi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 311587);