Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #150788 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#357 |
| commutator | C2×C16 | C22×C4 |
| Frattini, cosocle | C22×C16 | C23 |
| prime radical | #33733 | C2 |
| LCS | G ▷ C2×C16 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C2×C4×C8 ◁ C2×C4×C16 ◁ G | |
| LECS | G ▷ C22×C16 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
229 conjugacy classes of subgroups; only the 72 normal subgroups are shown.
This group has 229 conjugacy classes of subgroups — too many to draw. Only its 72 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 3 generators on 48 points:
(1,2)(3,9,11,6)(4,7)(5,8)(10,15,12,14)(13,16)(17,18,20,28,24,31,23,42)(19,32,35,48,38,25,37,43)(21,40,34,30,22,41,39,29)(26,46,45,36,27,47,44,33);(1,3)(2,6)(4,12)(5,11)(7,15)(8,9)(10,13)(14,16)(18,25)(19,33)(20,23)(21,39)(22,34)(26,29)(27,30)(28,48)(31,32)(35,47)(36,38)(37,46)(40,44)(41,45)(42,43);(1,4,5,13)(2,7,8,16)(3,10,11,12)(6,14,9,15)(17,19,34,47,23,37,21,36,24,38,39,46,20,35,22,33)(18,26,41,48,28,45,29,25,31,27,40,43,42,44,30,32)Polycyclic presentation:
2^9|a^2=e,c^2=fhi,d^2=gh,e^2=i,f^2=hi,g^2=hi,h^2=i,b^a=bd,c^a=cg,d^a=df,f^a=fh,g^a=gh,h^a=hi,c^b=cfi,d^b=dg,e^b=efgh,f^b=fh,g^b=gh,h^b=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 57991);