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×D4 | C23 |
| socle | C22 | C4×#99 |
| prime radical | C22×D4 | #41 |
| LCS | G ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C8 ◁ C2×C8×D4 ◁ G | |
| LECS | G ▷ C2×C4×D4 ▷ C23 ▷ C2 ▷ C1 | |
371 conjugacy classes of subgroups; only the 90 normal subgroups are shown.
This group has 371 conjugacy classes of subgroups — too many to draw. Only its 90 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)(17,18)(19,26)(20,24)(21,25)(22,27)(23,30)(28,31)(29,32);(2,7,10,9)(8,14,16,15)(17,19)(18,22)(20,29,30,27)(21,28)(23,26,24,31)(25,32);(1,3,5,12,6,13,4,11)(2,8,9,15,10,16,7,14)(17,20)(18,23)(19,27)(21,30)(22,31)(24,25)(26,32)(28,29)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=e,d^2=g,e^2=g,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, 36181);