Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #13584 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | C2×#353 |
| commutator | C42 | C22×C8 |
| Frattini, cosocle | #423 | C23 |
| socle | C2 | C2×#1318 |
| prime radical | C2×#532 | C4 |
| LCS | G ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C23×C4 ◁ C2×#423 ◁ G | |
| LECS | G ▷ #423 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
483 conjugacy classes of subgroups; only the 80 normal subgroups are shown.
This group has 483 conjugacy classes of subgroups — too many to draw. Only its 80 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,4,9,5,10,19,27,6,11,20,28,21,8,18,25)(3,12,14,26,16,22,30,31,13,24,29,32,15,7,17,23);(1,3)(2,7)(4,17)(5,16)(6,13)(8,24)(9,26)(10,12)(11,22)(14,19)(15,21)(18,29)(20,30)(23,25)(27,31)(28,32);(3,13)(7,22)(12,24)(14,29)(15,16)(17,30)(23,31)(26,32)Polycyclic presentation:
2^9|a^2=e,d^2=hi,e^2=g,f^2=i,g^2=i,b^a=bd,d^a=df,f^a=fh,h^a=hi,c^b=ci,d^b=dhi,e^b=efh,f^b=fi,e^d=ehi,f^e=fi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72338323, #72338325, #72338361, … |
| maximal normal subgroup of | #72338323, #72338325, #72338361, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 56776);