Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2184 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C2×D8 |
| commutator | C2×C8 | C22×C8 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C23 | C2×#102 |
| prime radical | C24 | #102 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C22×C42 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
496 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)(6,9,11,12)(13,14,16,22,18,24,26,20,19,25,15,21,17,23,27,28);(5,6,10,9,8,11,7,12)(14,20)(15,26)(16,27)(17,18)(21,23)(22,24)(25,28);(2,4)Polycyclic presentation:
2^9|a^2=f,b^2=dfgi,d^2=gi,f^2=h,g^2=i,h^2=i,b^a=bd,c^a=ce,d^b=dg,f^b=fg,g^b=gi,h^b=hi116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 43801);