Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #135930 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1864 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | C22×D8 | C2×C4 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ #1450 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
305 conjugacy classes of subgroups; only the 92 normal subgroups are shown.
This group has 305 conjugacy classes of subgroups — too many to draw. Only its 92 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:
(1,2,5,3,4,6,8,7)(9,10,13,19,15,20,14,32)(11,21,25,36,22,33,26,40)(12,27,29,37,31,17,30,18)(16,28,34,23,35,39,38,24);(1,3,5,6,4,7,8,2)(9,11)(10,16,20,35)(12,28)(13,26)(14,25)(15,22)(17,33,27,21)(18,36,37,40)(19,38,32,34)(23,30)(24,29)(31,39);(2,6)(3,7)(10,17)(11,22)(12,29)(13,14)(16,21)(18,32)(19,37)(20,27)(23,28)(24,39)(30,31)(33,35)(34,40)(36,38)Polycyclic presentation:
2^9|a^2=f,b^2=fg,d^2=g,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=ch,d^b=dg,e^b=egi,f^b=fi,g^b=gi,d^c=dgi,e^c=egi,f^c=fi,g^c=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72319899, #72319904, #72319964, … |
| maximal normal subgroup of | #72319899, #72319904, #72319964, … |
| maximal quotient of | C2×#135930 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 43126);