Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #145718 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2044 |
| commutator, Frattini, cosocle | C2×C4×D4 | C23 |
| prime radical | C22×D4 | C2×D4 |
| LCS | G ▷ C2×C4×D4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ G | |
| LECS | G ▷ C2×C4×D4 ▷ C23 ▷ C2 ▷ C1 | |
360 conjugacy classes of subgroups; only the 62 normal subgroups are shown.
This group has 360 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 — 3 generators on 24 points:
(1,2,5,7)(3,8,6,4)(9,10,14,11)(12,19,21,24)(13,16,22,18)(15,23,17,20);(2,3)(4,8)(6,7)(9,11,14,10)(12,17,21,15)(13,16,22,18)(19,20,24,23);(1,3,4,2,5,6,8,7)(9,12)(10,15)(11,17)(13,20)(14,21)(16,19)(18,24)(22,23)Polycyclic presentation:
2^9|a^2=gi,b^2=i,c^2=d,d^2=g,e^2=g,f^2=g,b^a=bd,c^a=ce,d^a=dg,e^a=eg,f^a=fhi,c^b=cf,d^b=dg,e^b=eh,f^b=fg,h^c=hi,f^e=fi47 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 52947);