Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #142050 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1993 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C23×C8 | C23 |
| prime radical | C24 | AGL(1,ℤ/8) |
| LCS | G ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
345 conjugacy classes of subgroups; only the 82 normal subgroups are shown.
This group has 345 conjugacy classes of subgroups — too many to draw. Only its 82 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)(3,5,7,8)(4,6)(9,10)(11,20)(12,26,30,37)(13,24)(14,18)(15,23)(16,19)(17,29,34,31)(21,25)(22,28,38,40)(27,36,39,35)(32,33);(1,3,4,7)(2,5)(6,8)(9,11,21,10,15,24,13,18,16,25,20,19,14,33,32,23)(12,27,35,37,31,38,40,17,30,39,36,26,29,22,28,34);(2,6)(3,7)(9,12)(10,17,19,34)(11,22,25,38)(13,28)(14,29)(15,31)(16,30)(18,26,23,37)(20,35)(21,36)(24,27,33,39)(32,40)Polycyclic presentation:
2^9|a^2=f,b^2=def,c^2=eg,d^2=gh,e^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gi,c^b=cg,d^b=dh,e^b=ei,f^b=fhi,f^c=fi50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 49246);