Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11757 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#99 |
| commutator | C22×C4 | C2×C42 |
| Frattini, cosocle | C2×C4×Q8 | C23 |
| socle | C22 | C4×#99 |
| prime radical | C23 | C4×C22⋊C4 |
| LCS | G ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C8 ◁ C2×C8×Q8 ◁ G | |
| LECS | G ▷ C2×C4×Q8 ▷ C23 ▷ C2 ▷ C1 | |
334 conjugacy classes of subgroups; only the 92 normal subgroups are shown.
This group has 334 conjugacy classes of subgroups — too many to draw. Only its 92 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:
(1,2,5,9,6,10,4,7)(3,8,12,15,13,16,11,14)(18,23)(19,21)(20,28)(24,32)(25,29)(26,31);(2,7,10,9)(8,14,16,15)(17,18,21,24,22,25,19,26)(20,29,27,31,28,23,30,32);(1,3,5,12,6,13,4,11)(2,8,9,15,10,16,7,14)(17,19)(18,24)(20,27)(21,22)(23,31)(25,26)(28,30)(29,32)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=e,d^2=g,e^2=gi,f^2=i,b^a=bd,c^a=ci,d^a=dg,f^a=fgh,c^b=ch,d^b=dh,h^b=hi,f^c=fi,f^d=fi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 36193);