Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #1406 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #353 |
| commutator | C42 | C22×C8 |
| Frattini, cosocle | #382 | C23 |
| socle | C22 | C2×#353 |
| prime radical | C22×D4 | C22⋊C4 |
| LCS | G ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C4 ◁ C2×C42 ◁ C4×#124 ◁ G | |
| LECS | G ▷ #382 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
363 conjugacy classes of subgroups; only the 72 normal subgroups are shown.
This group has 363 conjugacy classes of subgroups — too many to draw. Only its 72 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) = 128 |
A minimal faithful permutation representation — 3 generators on 40 points:
(1,2,4,7,5,8,3,6)(9,10)(11,19,21,34)(12,23,24,39)(13,26,28,17)(14,18)(15,20,29,36)(16,25)(22,32,27,35)(30,37,31,38)(33,40);(9,11,14,22)(10,15,18,31)(12,20,24,38)(13,27,28,21)(16,32,33,19)(17,29,26,30)(23,35,39,34)(25,36,40,37);(1,3,5,4)(2,6,8,7)Polycyclic presentation:
2^9|a^2=e,b^2=i,c^2=g,d^2=h,e^2=g,f^2=i,b^a=bd,d^a=df,f^a=fh,h^a=hi,d^b=dh,e^b=efhi,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, 56882);