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 | C22×#41 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1199 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 28 points:
(1,2,3,4)(13,14,18,24,20,25,19,23)(15,26,27,17,21,16,22,28);(5,6)(7,10)(8,9)(11,12)(13,15,18,27,20,21,19,22)(14,17,23,26,25,28,24,16);(2,4)(6,9)(10,12)(13,16,20,26)(14,21,25,15)(17,19,28,18)(22,24,27,23);(5,7,8,11)(6,10,9,12)(13,17)(14,22)(15,23)(16,19)(18,26)(20,28)(21,24)(25,27)Polycyclic presentation:
2^9|a^2=f,b^2=e,c^2=i,d^2=g,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,f^c=fi,h^c=hi,f^d=fi,h^d=hi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 311601);