Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12952 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1864 |
| 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 | |
325 conjugacy classes of subgroups; only the 104 normal subgroups are shown.
This group has 325 conjugacy classes of subgroups — too many to draw. Only its 104 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)(11,21,24,36,26,33,25,40)(12,27,29,37,31,17,30,18)(16,28,34,22,35,39,38,23);(1,3,5,6,4,7,8,2)(9,11)(10,16,20,35)(12,28)(13,25)(14,24)(15,26)(17,33,27,21)(18,36,37,40)(19,38,32,34)(22,30)(23,29)(31,39);(10,17)(12,29)(13,14)(16,33)(18,32)(19,37)(20,27)(21,35)(22,39)(23,28)(24,25)(30,31)(34,36)(38,40)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,d^b=dg,e^b=egi,f^b=fi,g^b=gi,d^c=dgi,e^c=egi,f^c=fi,g^c=gi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72216436, #72216443, #72216456, … |
| maximal normal subgroup of | #72216436, #72216443, #72216456, … |
| maximal quotient of | #72225256, C2×#135906 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 43102);