Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #133267 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #8522 |
| commutator | C2×C42 | C22×C4 |
| Frattini, cosocle | C4×M16 | C23 |
| prime radical | #3498 | C4 |
| LCS | G ▷ C2×C42 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C4×M16 ◁ G | |
| LECS | G ▷ C4×M16 ▷ C2×C4 ▷ C2 ▷ C1 | |
612 conjugacy classes of subgroups; only the 76 normal subgroups are shown.
This group has 612 conjugacy classes of subgroups — too many to draw. Only its 76 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) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,5,11,6,8,21,23,7,12,20,28,4,10,18,27)(3,13,16,29,15,9,24,25,14,22,19,31,17,26,30,32);(2,8)(4,6)(5,18)(9,22)(10,12)(11,28)(13,26)(15,17)(16,30)(19,24)(20,21)(25,32);(1,3)(2,9)(4,17)(5,19)(6,15)(7,14)(8,22)(10,13)(11,25)(12,26)(16,20)(18,24)(21,30)(23,31)(27,29)(28,32)Polycyclic presentation:
2^9|a^2=f,d^2=h,e^2=i,f^2=g,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,h^a=hi,d^b=dh,e^b=ei,f^b=fghi,g^b=gi,d^c=di,e^c=ei,f^c=fhi,f^d=fi,f^e=fi38 conjugacy classes — view the full character table.
| maximal subgroup of | #72339826, #72339846, #72339847, … |
| maximal normal subgroup of | #72339826, #72339846, #72339847, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 40463);