Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11753 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#99 |
| commutator | C22×C4 | C22×C8 |
| Frattini, cosocle | C2×C4⋊C8 | C23 |
| socle | C22 | C2×#324 |
| prime radical | C24 | C22⋊C8 |
| LCS | G ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ C2×#431 ◁ G | |
| LECS | G ▷ C2×C4⋊C8 ▷ C22×C4 ▷ C22 ▷ C1 | |
285 conjugacy classes of subgroups; only the 80 normal subgroups are shown.
This group has 285 conjugacy classes of subgroups — too many to draw. Only its 80 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 48 points:
(1,2,5,10,6,11,3,8)(4,7,12,15,13,9,14,16)(17,18,21,26,22,27,36,44,23,28,37,45,20,25,34,42)(19,29,32,46,31,39,47,43,30,41,35,48,33,24,38,40);(2,7,11,9)(3,5)(8,16,10,15)(12,14)(17,19)(18,24)(20,33)(21,35)(22,31)(23,30)(25,41)(26,43)(27,29)(28,39)(32,37)(34,47)(36,38)(40,45)(42,46)(44,48);(1,3,6,5)(2,8,11,10)(4,12,13,14)(7,15,9,16)(19,30)(24,39)(29,41)(31,33)(32,35)(38,47)(40,43)(46,48)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=h,d^2=hi,e^2=g,g^2=i,b^a=bd,d^a=df,f^a=fh,c^b=chi,d^b=di,e^b=efhi,e^d=eh74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 29538);