Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13247 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1972 |
| commutator | C2×C8 | C22×C8 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | C22×D16 | C4 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ #1450 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
348 conjugacy classes of subgroups; only the 102 normal subgroups are shown.
This group has 348 conjugacy classes of subgroups — too many to draw. Only its 102 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:
(1,2,4,7,5,3,6,8)(9,10)(11,13)(12,22,28,38)(14,23)(15,32)(16,20)(17,27,34,36)(18,21)(19,30)(24,31)(25,29,33,39)(26,35,37,40);(1,3,4,8,5,2,6,7)(9,11,21,19,15,23,31,20,16,24,14,32,30,18,13,10)(12,25,27,17,29,38,26,35,28,33,36,34,39,22,37,40);(9,12,21,36,15,29,31,37,16,28,14,27,30,39,13,26)(10,17,24,38,19,35,18,33,20,34,11,22,32,40,23,25)Polycyclic presentation:
2^9|a^2=f,b^2=dfg,c^2=ei,d^2=gi,e^2=g,f^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eg,g^a=gi,c^b=cg,e^b=ei,f^c=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72216421, #72216444, #72216451, … |
| maximal normal subgroup of | #72216421, #72216444, #72216451, … |
| maximal quotient of | #72227094 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 47943);