Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #7076 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #327 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C2×C4⋊C8 | C23 |
| socle | C22 | C2×#327 |
| prime radical | C24 | #102 |
| LCS | G ▷ C22×C8 ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ C2×#431 ◁ G | |
| LECS | G ▷ C2×C4⋊C8 ▷ C22×C4 ▷ C22 ▷ C1 | |
337 conjugacy classes of subgroups; only the 72 normal subgroups are shown.
This group has 337 conjugacy classes of subgroups — too many to draw. Only its 72 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:
(1,2,4,8,5,6,11,13)(3,7,9,14,10,12,15,16)(17,18,21,26,23,27,22,38)(19,28,30,42,32,45,31,48)(20,33,35,25,37,43,36,44)(24,34,39,47,41,46,40,29);(2,6)(7,12)(17,19,23,32)(18,24,27,41)(20,34,37,46)(21,31,22,30)(25,42,44,48)(26,39,38,40)(28,43,45,33)(29,36,47,35);(1,3,5,10)(2,7,6,12)(4,9,11,15)(8,14,13,16)Polycyclic presentation:
2^9|a^2=e,b^2=i,c^2=h,d^2=gi,e^2=hi,f^2=i,g^2=i,b^a=bd,d^a=df,f^a=fh,g^a=gi,d^b=dg,e^b=efgi,f^b=fi,g^b=gi,e^d=ehi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 29855);