Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #13661 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×D16 |
| commutator | C2×C16 | C22×C4 |
| Frattini, cosocle | C4×C16 | C23 |
| socle | C22 | C4×D16 |
| prime radical | C2×D32 | C4 |
| LCS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C4×C8 ◁ C82 ◁ C8×C16 ◁ G | |
| LECS | G ▷ C4×C16 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
149 conjugacy classes of subgroups; only the 56 normal subgroups are shown.
This group has 149 conjugacy classes of subgroups — too many to draw. Only its 56 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 3 generators on 40 points:
(1,2,4,3,5,6,8,7)(9,10,13,19)(11,17,15,29)(12,16,25,31)(14,28,27,40)(18,35,36,23)(20,22,37,24)(21,30,26,33)(32,39,34,38);(10,16)(11,15)(12,23)(14,24)(17,33)(18,20)(19,31)(21,38)(22,27)(25,35)(26,39)(28,32)(29,30)(34,40)(36,37);(1,3,4,6,5,7,8,2)(9,11,13,15)(10,17,19,29)(12,21,25,26)(14,22,27,24)(16,30,31,33)(18,34,36,32)(20,28,37,40)(23,39,35,38)Polycyclic presentation:
2^9|a^2=e,c^2=e,d^2=fh,e^2=g,f^2=hi,h^2=i,b^a=bd,c^a=cg,d^a=df,f^a=fh,h^a=hi,c^b=ci,d^b=df,f^b=fh,h^b=hi98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 58307);