Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1999 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C8 | C2×D4 |
| commutator | C22×C4 | C22×C8 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C23 | C2×C4⋊C8 |
| prime radical | C24 | C4⋊C8 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C8 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
372 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)(5,6,8,11)(7,9,12,10)(13,14,18,15,19,21,28,24,17,20,22,23,16,25,27,26);(6,9)(7,12)(10,11)(13,15,22,25,19,24,27,14,17,23,18,21,16,26,28,20);(2,4)(14,20)(15,23)(21,25)(24,26)Polycyclic presentation:
2^9|a^2=f,b^2=fg,d^2=g,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dg,c^b=ci,d^b=dg128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 43020);