Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #93508 |
| automorphism | ? |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | H(3,ℤ/4) |
| commutator | C22×C8 | C42 |
| Frattini, cosocle | C2×C4×M16 | C22 |
| socle | C22 | #1288 |
| prime radical | C24 | #104 |
| LCS | G ▷ C22×C8 ▷ C2×C4 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C8 ◁ G | |
| LECS | G ▷ C2×C4×M16 ▷ C22×C4 ▷ C2 ▷ C1 | |
326 conjugacy classes of subgroups; only the 64 normal subgroups are shown.
This group has 326 conjugacy classes of subgroups — too many to draw. Only its 64 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) = 32 |
A minimal faithful permutation representation — 2 generators on 24 points:
(2,6)(3,4)(7,8)(9,10)(11,21,17,24)(12,18,13,23)(14,19)(15,20)(16,22);(1,2,4,7,5,8,3,6)(9,11,13,20,14,23,21,22,15,17,12,10,16,18,24,19)Polycyclic presentation:
2^9|a^2=d,b^2=e,c^2=fhi,e^2=f,f^2=i,g^2=i,b^a=bc,c^a=cf,e^a=efghi,h^a=hi,c^b=cg,d^b=dh,h^b=hi,d^c=di,e^c=ei,e^d=ei68 conjugacy classes — view the full character table.
| maximal subgroup of | #72338350, #72338368, #72338369, … |
| maximal normal subgroup of | #72338350, #72338368, #72338369, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 751);