Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2184 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C2×D8 |
| commutator | C2×C8 | C2×C42 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle, prime radical | C23 | C4×D8 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C2×C4×C8 ◁ C2×C82 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
338 conjugacy classes of subgroups; only the 134 normal subgroups are shown.
This group has 338 conjugacy classes of subgroups — too many to draw. Only its 134 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 3 generators on 28 points:
(1,2)(3,4)(5,6,9,7,8,10,12,11)(13,14)(15,25)(16,20)(17,22)(18,21)(19,24)(23,27)(26,28);(2,4)(6,10)(7,11)(13,15,18,20,16,21,25,14,19,22,28,23,27,26,17,24);(5,7,9,10,8,11,12,6)(13,16,19,27)(14,20,24,23)(15,26,22,21)(17,18,25,28)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=ei,d^2=gi,e^2=h,f^2=g,g^2=i,b^a=bd,c^a=ch,d^a=dg,f^a=fgi,g^a=gi,c^b=chi116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 35943);