Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of AGL(1,ℤ/8) |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C82 · Schur multiplier (H2) = C22×C8 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4×C8 | C2×D4 |
| commutator | C4 | C2×C82 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | C8×C22⋊C4 |
| prime radical | C2×#41 | C42 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C8 ◁ C2×C4×C16 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
402 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 3 generators on 40 points:
(9,10,13,19,14,18,26,33,15,20,27,34,12,16,25,29)(11,17,23,31,24,30,37,39,21,32,35,40,22,28,36,38);(1,2,3,5,4,6,7,8)(10,16,20,18)(11,21)(17,30,32,28)(19,29,34,33)(22,24)(23,35)(31,39,40,38)(36,37);(9,11,13,23,14,24,26,37,15,21,27,35,12,22,25,36)(10,17,19,31,18,30,33,39,20,32,34,40,16,28,29,38)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=e,d^2=i,e^2=g,f^2=h,g^2=i,b^a=bd,d^a=di,f^a=fi,c^b=ci176 conjugacy classes — view the full character table.
| maximal subgroup of | C8×#2171, C2×#128250 |
| maximal normal subgroup of | C8×#2171, C2×#128250 |
| maximal quotient of | C2×#128250 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 35446);