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 | C23×D4 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
981 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 28 points:
(2,4)(5,6,9,8)(7,10,11,12)(14,18)(15,21)(16,24)(19,25)(22,23)(26,28);(5,7)(6,10)(8,12)(9,11)(13,14,16,18,17,19,24,25)(15,20,22,26,23,27,21,28);(1,2,3,4)(7,11)(10,12)(14,19)(18,25)(20,27)(26,28);(5,8,9,6)(7,12,11,10)(13,15,17,23)(14,20,19,27)(16,22,24,21)(18,26,25,28)Polycyclic presentation:
2^9|a^2=gi,b^2=e,c^2=f,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, 312260);