Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #476095 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C2×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | H(3,𝔽4) |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #2804 |
| prime radical | C2×#532 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
833 conjugacy classes of subgroups; only the 154 normal subgroups are shown.
This group has 833 conjugacy classes of subgroups — too many to draw. Only its 154 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,4,6)(3,5,7,8)(9,10,15,17,14,11,16,20)(12,18,24,23,19,13,22,21);(1,3)(2,5)(4,7)(6,8)(9,11,15,20,14,10,16,17)(12,21,24,18,19,23,22,13);(3,7)(5,8)(9,12,14,19)(10,13)(11,18)(15,22,16,24)(17,23)(20,21);(9,13,14,18)(10,12)(11,19)(15,23,16,21)(17,22)(20,24)Polycyclic presentation:
2^9|a^2=eg,b^2=eh,c^2=ef,d^2=f,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,c^b=cg,d^b=def,f^b=fi,d^c=de,e^c=ei,f^c=fi,e^d=ei56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 387027);