Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12019 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | C2×#106 |
| commutator | C2×C8 | C22×C8 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C43 ◁ C42×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
300 conjugacy classes of subgroups; only the 178 normal subgroups are shown.
This group has 300 conjugacy classes of subgroups — too many to draw. Only its 178 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:
(2,4)(5,6,7,9,8,10,11,12)(14,18,19,24)(15,21)(16,23)(20,27,26,25)(22,28);(13,14,15,18,16,20,21,25,17,19,22,24,23,26,28,27);(1,2,3,4)Polycyclic presentation:
2^9|a^2=f,b^2=d,c^2=e,d^2=g,f^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gi,f^b=fi104 conjugacy classes — view the full character table.
| maximal subgroup of | #72216168, #72216172, #72216174, … |
| maximal normal subgroup of | #72216168, #72216172, #72216174, … |
| maximal quotient of | #72216168, #72225128, #72225131, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 42911);