Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11669 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #418 |
| commutator | C2×C8 | C2×C42 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | #1490 |
| prime radical | C2×#135 | C2×C4 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C8 ◁ C4×#97 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
431 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 20 points:
(1,2,4,3)(5,6)(7,10)(8,17)(9,13)(11,16)(12,15)(14,18)(19,20);(6,10,12,19,13,18,17,11);(1,3,4,2)(7,14)(10,18)(11,19)(16,20)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=e,d^2=gi,f^2=h,g^2=i,b^a=bd,d^a=dg,f^a=fgi,g^a=gi,h^a=hi,c^b=ch,d^c=di92 conjugacy classes — view the full character table.
| maximal subgroup of | #72213074, #72213969, #72214092, … |
| maximal normal subgroup of | #72213969, #72214092, #72214124, … |
| maximal quotient of | #72213035, #72213037, #72213038, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 35766);