Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13156 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1936 |
| commutator | C2×C8 | C22×C8 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | C22×D8 | C2×C4 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ #1450 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
261 conjugacy classes of subgroups; only the 90 normal subgroups are shown.
This group has 261 conjugacy classes of subgroups — too many to draw. Only its 90 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)(10,16,19,27)(11,21,24,30)(12,26)(13,28)(14,31)(17,33)(18,35,29,20)(23,37,38,25)(32,36)(34,39);(1,3,4,8,5,2,6,7)(9,10,13,29,31,35,28,16,15,19,26,18,14,20,12,27)(11,22,21,32,37,17,23,36,24,40,30,39,25,33,38,34);(9,11,12,23,14,25,26,21,15,24,28,38,31,37,13,30)(10,17,16,32,20,22,29,34,19,33,27,39,35,40,18,36)Polycyclic presentation:
2^9|a^2=f,b^2=dfgi,c^2=d,d^2=g,e^2=gi,f^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=egi,g^a=gi,c^b=cg,d^b=di,e^b=ei,f^b=fi,e^c=ei,f^c=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72216436, #72216448, #72216451, … |
| maximal normal subgroup of | #72216436, #72216448, #72216451, … |
| maximal quotient of | #72227054 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 47386);