Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2218 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C8 | C2×D8 |
| commutator | C2×C8 | C22×C8 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | C2×D4⋊C8 |
| prime radical | #3147 | C4 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8 ◁ C2×C4×C8 ◁ C42×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
442 conjugacy classes of subgroups; only the 130 normal subgroups are shown.
This group has 442 conjugacy classes of subgroups — too many to draw. Only its 130 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(2,4)(5,6,9,14,10,16,8,24,11,17,25,13,19,15,26,31)(7,18,22,29,23,33,21,12,20,28,36,30,34,32,35,27);(5,7,19,34,11,20,10,23)(6,12,17,27)(8,21,9,22,26,35,25,36)(13,18,14,28)(15,29,16,30)(24,32,31,33);(1,2,3,4)(7,20)(12,27)(18,28)(21,35)(22,36)(23,34)(29,30)(32,33)Polycyclic presentation:
2^9|a^2=f,b^2=dfg,c^2=e,d^2=gi,f^2=h,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gi,c^b=ci104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 47728);