Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #133394 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1790 |
| commutator | C22×D4 | C22×C4 |
| Frattini, cosocle | C2×C4×D4 | C23 |
| prime radical | C2×#120 | C2×C4 |
| LCS | G ▷ C22×D4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ G | |
| LECS | G ▷ C2×C4×D4 ▷ C23 ▷ C2 ▷ C1 | |
652 conjugacy classes of subgroups; only the 78 normal subgroups are shown.
This group has 652 conjugacy classes of subgroups — too many to draw. Only its 78 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:
(1,2,5,10,6,9,4,8)(3,11,12,16,13,7,14,15)(17,18)(19,25)(20,28)(21,24)(22,27)(23,30)(26,32)(29,31);(1,3)(2,7)(4,12)(5,14)(6,13)(8,15)(9,11)(10,16)(17,19,21,26)(18,22,24,31)(20,29,30,27)(23,32,28,25);(1,4,6,5)(2,8,9,10)(3,12,13,14)(7,15,11,16)(19,26)(20,30)(22,31)(25,32)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=gi,d^2=i,f^2=gi,b^a=bd,c^a=ce,d^a=dg,h^a=hi,c^b=ci,d^b=di,e^b=ehi,f^b=fgi,d^c=dh,e^d=ei44 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 40885);