Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #99 |
| automorphism | ? |
| homology | Abelianization (H1) = C8×C16 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4×C8 | C22⋊C4 |
| commutator | C22 | C8×C16 |
| Frattini, cosocle | C2×C82 | C22 |
| socle | C22 | #1273 |
| prime radical | C23 | C82 |
| LCS | G ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C8 ◁ C2×C82 ◁ G | |
| LECS | G ▷ C2×C82 ▷ C42 ▷ C22 ▷ C1 | |
198 conjugacy classes of subgroups; only the 122 normal subgroups are shown.
This group has 198 conjugacy classes of subgroups — too many to draw. Only its 122 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:
(1,2,3,7,4,8,10,14,5,6,9,12,11,13,15,16);(2,6)(3,9)(8,13)(10,15)(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,d^2=f,e^2=g,f^2=h,g^2=i,b^a=bc,c^a=ch,d^b=dh176 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 94);