Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #393909 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C23×D8 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C4×#95 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2328 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 20 points:
(2,4)(6,8)(13,14,15,17)(16,18,20,19);(1,2,3,4)(9,10)(11,12)(14,17)(18,19);(5,6,7,8)(10,12);(14,18)(16,20)(17,19)Polycyclic presentation:
2^9|a^2=i,b^2=ei,c^2=f,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,h^d=hi86 conjugacy classes — view the full character table.
| maximal subgroup of | #72265166, #72265178, #72265274 |
| maximal normal subgroup of | #72265166, #72265178, #72265274 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 305078);