Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11541 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#99 |
| commutator | C24 | C2×C42 |
| Frattini, cosocle | C22×C22⋊C4 | C23 |
| prime radical | C23×D4 | C2×C4 |
| LCS | G ▷ C24 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ C22×C4×D4 ◁ G | |
| LECS | G ▷ C22×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
1000 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 20 points:
(5,6,8,12,9,10,7,11)(13,14)(15,19,20,17)(16,18);(1,2,3,4)(6,10)(7,8)(14,17)(18,19);(5,7,9,8)(6,11,10,12)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=h,e^2=h,b^a=bd,d^a=dg,g^a=gi,c^b=ch,e^b=eg,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 34209);