Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13304 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C2×D8 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C23 | C2×#107 |
| prime radical | C24 | #107 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C22×C42 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
358 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)(5,6,9,7,8,10,12,11)(13,14,16,20)(15,18,24,26)(17,25,23,21)(19,28,27,22);(5,7,9,10,8,11,12,6)(14,18)(15,22)(17,23)(19,21)(20,26)(24,28)(25,27);(2,4)(6,10)(7,11)Polycyclic presentation:
2^9|a^2=f,b^2=fi,d^2=gi,f^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gi,c^b=ch,d^b=dg,g^b=gi104 conjugacy classes — view the full character table.
| maximal subgroup of | #72321045, #72323077, #72323099 |
| maximal normal subgroup of | #72321045, #72323077, #72323099 |
| maximal quotient of | #72321083 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 43036);