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 | C8×D4 |
| prime radical | C22×D4 | C2×C8 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C8 ◁ C2×C4×C16 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
401 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 3 generators on 28 points:
(2,4)(5,6,8,10,9,12,7,11)(13,14,17,20,18,21,25,27,16,15,19,22,24,23,26,28);(1,2,3,4)(6,10)(7,8)(11,12)(13,15,17,22,18,23,25,28,16,14,19,20,24,21,26,27);(6,10)(7,8)(11,12)(14,15)(20,22)(21,23)(27,28)Polycyclic presentation:
2^9|a^2=f,b^2=df,d^2=g,e^2=g,f^2=h,h^2=i,b^a=bd,c^a=ce,c^b=ci,d^b=dg,e^b=eg,f^b=fg,d^c=dg,e^c=eg,f^c=fg128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 43844);