Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #7402 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #353 |
| commutator | C2×C42 | C22×C4 |
| Frattini, cosocle | #390 | C23 |
| socle | C22 | C2×#353 |
| prime radical | C2×#127 | D4 |
| LCS | G ▷ C2×C42 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C4 ◁ C2×C42 ◁ C4×#127 ◁ G | |
| LECS | G ▷ #390 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
405 conjugacy classes of subgroups; only the 58 normal subgroups are shown.
This group has 405 conjugacy classes of subgroups — too many to draw. Only its 58 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) = 64 |
A minimal faithful permutation representation — 3 generators on 24 points:
(2,6,7,8)(9,10)(11,18,19,14)(12,16,22,15)(13,17)(20,23,21,24);(1,2)(3,6)(4,8)(5,7)(9,11,13,20)(10,14,17,24)(12,21,22,19)(15,23,16,18);(1,3,5,4)(2,6,7,8)Polycyclic presentation:
2^9|a^2=e,b^2=i,c^2=ghi,d^2=g,f^2=i,b^a=bd,d^a=df,f^a=fh,g^a=gi,h^a=hi,d^b=dg,e^b=efgi,f^b=fi,e^d=ehi,f^e=fi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 57527);