Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #488889 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#447 |
| commutator, Frattini, cosocle | C4×C8 | C24 |
| prime radical | #33580 | C2 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C2×C8×Q8 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
842 conjugacy classes of subgroups; only the 160 normal subgroups are shown.
This group has 842 conjugacy classes of subgroups — too many to draw. Only its 160 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)(9,10)(11,14)(12,21)(13,15)(16,18)(17,19)(20,22)(23,24);(1,2,4,6)(3,8,7,5)(10,14)(11,18)(12,19)(15,22)(16,17)(20,24)(21,23);(10,14)(11,18)(12,19)(15,22)(16,17)(20,24)(21,23);(10,15)(14,22)(16,23)(17,21)Polycyclic presentation:
2^9|b^2=ghi,e^2=gi,f^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=di,e^a=eg,f^a=fh,g^a=gi,h^a=hi,e^b=eg,f^b=fh,g^b=gi,h^b=hi,e^c=eh,f^c=fh,g^c=gi,h^c=hi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72342446, #72342447, #72342450, … |
| maximal normal subgroup of | #72342446, #72342447, #72342450, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 396232);