Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #124648 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #418 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | #1490 |
| prime radical | C2×#135 | C2×C4 |
| LCS | G ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C8 ◁ C4×#97 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C22×C4 ▷ C2 ▷ C1 | |
303 conjugacy classes of subgroups; only the 68 normal subgroups are shown.
This group has 303 conjugacy classes of subgroups — too many to draw. Only its 68 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) = 64 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2)(3,9,11,6)(4,7)(5,8)(10,15,16,13)(12,14)(18,22,23,31,25,30,24,26);(1,3)(2,6)(4,10)(5,11)(7,13)(8,9)(12,16)(14,15)(17,18)(19,26)(20,23)(21,25)(22,29)(24,28)(27,31)(30,32);(1,4,5,12)(2,7,8,14)(3,10,11,16)(6,13,9,15)(19,27)(22,30)(26,31)(29,32)Polycyclic presentation:
2^9|a^2=e,c^2=f,d^2=gi,e^2=h,g^2=i,b^a=bd,c^a=ch,d^a=df,d^b=dg,e^b=efg,g^b=gi,h^b=hi,d^c=di68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 31873);