Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #327 |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C16 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C42 | C22⋊C4 |
| commutator | C2×C4 | C4×C16 |
| Frattini, cosocle | C22×C4×C8 | C22 |
| socle | C23 | #336 |
| prime radical | C24 | C4⋊C8 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C42 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C22×C4×C8 ▷ C22×C4 ▷ C2 ▷ C1 | |
326 conjugacy classes of subgroups; only the 110 normal subgroups are shown.
This group has 326 conjugacy classes of subgroups — too many to draw. Only its 110 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 2 generators on 32 points:
(1,2,4,7,5,8,3,6)(9,10)(11,15)(12,14)(13,16);(2,6,8,7)(9,11)(10,13,14,15)(12,16)(17,18,19,22,20,23,25,28,21,24,26,29,27,30,31,32)Polycyclic presentation:
2^9|a^2=d,b^2=e,c^2=f,d^2=f,e^2=h,h^2=i,b^a=bc,c^a=cf,e^a=efg,c^b=cg152 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 592);