Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #146231 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2094 |
| commutator, Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | C23 | #492 |
| LCS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C2×#322 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
271 conjugacy classes of subgroups; only the 96 normal subgroups are shown.
This group has 271 conjugacy classes of subgroups — too many to draw. Only its 96 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:
(1,2,6,8,5,3,4,7)(11,18)(14,19)(16,24)(20,21);(1,3,5,2)(4,7,6,8)(9,10,12,22,13,15,23,17)(11,19,21,16,18,14,20,24);(1,2,5,3)(4,8,6,7)(10,14)(11,20)(12,23)(15,19)(16,22)(17,24)(18,21)Polycyclic presentation:
2^9|a^2=dg,b^2=deh,c^2=g,d^2=g,e^2=g,f^2=h,h^2=i,b^a=bd,c^a=ce,f^a=fi,c^b=cf,d^b=dg,e^b=egi,f^b=fh,h^b=hi,d^c=dg,e^c=eg,f^c=fhi,h^c=hi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 53473);