Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10012 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | #127 |
| commutator | C2×C42 | C22×C4 |
| Frattini, cosocle | C22×C42 | C23 |
| socle | C23 | #392 |
| prime radical | C2×#3150 | C2 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C42 ▷ C22 ▷ C1 | |
1262 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 3 generators on 28 points:
(1,2,3,4)(5,6)(7,10)(8,11)(9,12)(18,27)(21,28);(6,10)(7,8)(11,12)(13,14)(15,21)(16,24)(17,19)(18,23)(20,26)(22,28)(25,27);(6,11)(7,8)(10,12)(13,15)(14,18)(16,25)(17,22)(19,27)(20,28)(21,24)(23,26)Polycyclic presentation:
2^9|a^2=g,d^2=h,e^2=h,f^2=hi,b^a=bd,c^a=ce,d^a=dh,e^a=eh,c^b=cf,d^b=dh,e^b=ei,f^b=fhi,d^c=di,e^c=eh,f^c=fhi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72211651, C2×#111503, #72216744, … |
| maximal normal subgroup of | #72211651, C2×#111503, #72216744, … |
| maximal quotient of | #72211633, C2×#111503 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 18699);