Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #501040 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D8 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | C22×#493 | C2 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
1630 conjugacy classes of subgroups; only the 234 normal subgroups are shown.
This group has 1630 conjugacy classes of subgroups — too many to draw. Only its 234 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:
(2,3)(10,14)(11,18)(12,19)(15,20)(16,24)(17,23)(21,22);(1,2,4,3)(5,6)(7,8);(6,8)(9,10,15,24,12,16,11,14,13,17,18,22,19,21,20,23);(1,3,4,2)Polycyclic presentation:
2^9|b^2=e,c^2=fhi,d^2=e,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,h^a=hi,c^b=cg98 conjugacy classes — view the full character table.
| maximal subgroup of | #72309928, D4×#2187, #72310196 |
| maximal normal subgroup of | #72309928, D4×#2187, #72310196 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 410297);