Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of AGL(1,ℤ/8) |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C16 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C8 | C2×D4 |
| commutator | C4 | C2×C4×C16 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle, prime radical | C23 | C8×D4 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C8 ◁ C2×C4×C16 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
294 conjugacy classes of subgroups; only the 170 normal subgroups are shown.
This group has 294 conjugacy classes of subgroups — too many to draw. Only its 170 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:
(5,6,8,10,9,11,7,12)(13,14,16,20,17,21,24,27,18,15,19,22,25,23,26,28);(1,2,3,4)(6,10)(7,8)(11,12);(6,11)(10,12)(13,15,16,22,17,23,24,28,18,14,19,20,25,21,26,27)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=de,d^2=g,e^2=h,h^2=i,b^a=bd,c^a=cg,d^b=dg,e^b=eg176 conjugacy classes — view the full character table.
| maximal subgroup of | #72286540, #72286561, #72287193 |
| maximal normal subgroup of | #72286540, #72286561, #72287193 |
| maximal quotient of | #72286540, #72286543 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 36357);