Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #105681 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C4×C22⋊C4 |
| commutator | C22×C4 | C2×C42 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | #1749 |
| prime radical | C2×#41 | C42 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×#322 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
345 conjugacy classes of subgroups; only the 134 normal subgroups are shown.
This group has 345 conjugacy classes of subgroups — too many to draw. Only its 134 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)(11,19,21,15)(12,17)(13,18)(14,16)(20,29,23,27)(22,33,36,28)(24,30)(25,32)(26,31)(34,38)(35,40,37,39);(9,11,12,22,13,23,25,37,14,21,24,36,26,20,34,35)(10,15,17,28,18,29,32,40,16,19,30,33,31,27,38,39);(1,3,4,6,5,7,8,2)(11,20,21,23)(12,24)(15,27,19,29)(17,30)(22,37,36,35)(25,34)(28,40,33,39)(32,38)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=e,e^2=g,f^2=h,h^2=i,b^a=bd,c^a=cg,d^a=di,c^b=ch,e^b=ei,f^c=fi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72319890, #72319899, #72319950, … |
| maximal normal subgroup of | #72319890, #72319899, #72319950, … |
| maximal quotient of | C2×#128281 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 35477);