Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #491556 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#493 |
| commutator, Frattini, cosocle | C4×C8 | C24 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C8×M16 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
722 conjugacy classes of subgroups; only the 174 normal subgroups are shown.
This group has 722 conjugacy classes of subgroups — too many to draw. Only its 174 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)(10,14)(11,15)(12,18)(16,24)(17,22)(19,23)(20,21);(1,2,3,5,4,6,7,8);(9,10,15,21,12,16,19,22,13,17,23,24,18,20,11,14);(2,6)(5,8)Polycyclic presentation:
2^9|b^2=e,c^2=fh,e^2=g,f^2=hi,h^2=i,b^a=be,c^a=cf,e^a=eg,f^a=fh,h^a=hi,d^b=dg92 conjugacy classes — view the full character table.
| maximal subgroup of | #72342596, D16×AGL(1,ℤ/8), #72342610 |
| maximal normal subgroup of | #72342596, D16×AGL(1,ℤ/8), #72342610 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 398752);