Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #394177 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#466 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C23×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2322 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2)(3,4)(9,10)(11,12)(14,15)(16,17)(19,20);(5,6)(7,8)(13,14,17,19,18,20,16,15);(2,4)(6,8)(13,15,16,20,18,19,17,14);(10,12)Polycyclic presentation:
2^9|b^2=ei,c^2=e,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg104 conjugacy classes — view the full character table.
| maximal subgroup of | #72311280, #72311289, #72311365 |
| maximal normal subgroup of | #72311280, #72311289, #72311365 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 301931);