Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1284 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C8 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C42 | #95 |
| commutator | C2×C4 | C2×C4×C8 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | C2×#336 |
| prime radical | C24 | C22⋊C8 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C42 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
628 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2)(3,5)(4,7)(6,8)(9,10,11,15,13,16,18,22,12,14,17,20,19,21,23,24);(2,5,7,6)(10,14)(11,17)(16,21)(18,23);(10,14)(15,20)(16,21)(22,24)Polycyclic presentation:
2^9|a^2=e,b^2=f,d^2=gi,e^2=h,h^2=i,b^a=bd,c^a=ci,d^a=dg,f^a=fgi,e^b=ei128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 33822);