Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #394289 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
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 | |
1192 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 28 points:
(5,6,8,9)(7,10,11,12)(13,14)(15,16)(17,25)(18,22)(19,21)(20,23)(24,26)(27,28);(1,2)(3,4)(6,9)(10,12)(13,15,18,19)(14,16,22,21)(17,23,26,27)(20,24,28,25);(2,4)(13,16,19,22,18,21,15,14)(17,24,27,28,26,25,23,20);(5,7,8,11)(6,10,9,12)(13,17)(14,20)(15,23)(16,24)(18,26)(19,27)(21,25)(22,28)Polycyclic presentation:
2^9|a^2=e,b^2=i,c^2=fi,d^2=e,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,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, 305471);