Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #93011 |
| automorphism | ? |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1289 |
| commutator | C22×C8 | C42 |
| Frattini, cosocle | C2×C8×D4 | C22 |
| prime radical | C22×D4 | D8 |
| LCS | G ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C8×D4 ◁ G | |
| LECS | G ▷ C2×C8×D4 ▷ C22×C4 ▷ C22 ▷ C1 | |
255 conjugacy classes of subgroups; only the 62 normal subgroups are shown.
This group has 255 conjugacy classes of subgroups — too many to draw. Only its 62 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 — 2 generators on 24 points:
(1,2)(3,7,8,5)(4,6)(10,15)(11,19)(12,21)(13,20)(16,17)(18,24)(22,23);(2,5)(6,7)(9,10,12,16,13,17,11,15,14,18,19,23,20,22,21,24)Polycyclic presentation:
2^9|a^2=d,b^2=e,c^2=fghi,e^2=fg,f^2=hi,b^a=bc,c^a=cf,e^a=efghi,f^a=fh,g^a=gi,d^b=dg,d^c=di68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 207);