Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #13661 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×D16 |
| commutator | C2×C16 | C22×C4 |
| Frattini, cosocle | C4×C16 | C23 |
| socle | C22 | C2×#357 |
| prime radical | C23 | #357 |
| LCS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ C2×C4×C8 ◁ C2×C4×C16 ◁ G | |
| LECS | G ▷ C4×C16 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
206 conjugacy classes of subgroups; only the 80 normal subgroups are shown.
This group has 206 conjugacy classes of subgroups — too many to draw. Only its 80 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:
(1,2,4,7,5,8,11,14)(3,6,10,13,9,12,15,16)(18,23)(19,27)(20,29)(21,33)(24,39)(25,42)(26,38)(28,32)(30,46)(31,36)(34,40)(35,37)(41,47)(43,44)(45,48);(1,3)(2,6)(4,10)(5,9)(7,13)(8,12)(11,15)(14,16)(17,18,22,26)(19,28,31,45)(20,32,34,48)(21,35,33,25)(23,36,38,27)(24,40,41,29)(30,47,43,39)(37,44,42,46);(3,9)(6,12)(10,15)(13,16)Polycyclic presentation:
2^9|a^2=e,b^2=i,d^2=fh,e^2=g,f^2=hi,h^2=i,b^a=bd,d^a=df,f^a=fh,h^a=hi,c^b=cg,d^b=df,f^b=fh,h^b=hi98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 57314);