Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #490693 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#459 |
| commutator, Frattini, cosocle | C4×C8 | C24 |
| prime radical | #33582 | C2 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C2×C82 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
539 conjugacy classes of subgroups; only the 142 normal subgroups are shown.
This group has 539 conjugacy classes of subgroups — too many to draw. Only its 142 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) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,5)(3,7)(6,8)(10,13)(11,16)(12,15)(17,18)(19,21)(20,24)(22,23);(1,2,3,5,4,6,7,8)(9,10,15,18,14,17,12,13)(11,19,23,22,21,16,20,24);(2,6)(5,8)(9,11)(10,16)(12,20)(13,22)(14,21)(15,23)(17,19)(18,24);(9,12,14,15)(10,13,17,18)(11,20,21,23)(16,22,19,24)Polycyclic presentation:
2^9|b^2=ei,d^2=i,e^2=g,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=di,e^a=eg,f^a=fh,h^a=hi,c^b=cg,f^c=fh,h^c=hi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72309935, #72309937, #72309938, … |
| maximal normal subgroup of | #72309935, #72309937, #72309938, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 397893);