Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2127 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C16 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#99 |
| commutator | C2×C4 | C22×C16 |
| Frattini, cosocle | C2×C4⋊C8 | C23 |
| socle | C22 | C2×#324 |
| prime radical | C24 | C22⋊C8 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ C22×C4⋊C8 ◁ G | |
| LECS | G ▷ C2×C4⋊C8 ▷ C22×C4 ▷ C22 ▷ C1 | |
331 conjugacy classes of subgroups; only the 100 normal subgroups are shown.
This group has 331 conjugacy classes of subgroups — too many to draw. Only its 100 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) = 128 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2)(3,9,10,15)(4,12,13,7)(5,8)(6,11)(14,16)(17,18,19,22,20,23,25,28,21,24,26,29,27,30,31,32);(1,3)(2,6,8,14)(4,11)(5,10)(7,9,12,15)(13,16);(3,10)(6,14)(9,15)(11,16)Polycyclic presentation:
2^9|a^2=e,b^2=f,d^2=h,e^2=g,g^2=i,b^a=bd,d^a=df,f^a=fh,c^b=ch,e^b=efh,e^d=eh92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 29493);