Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #149901 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×AΣL(1,𝔽16) |
| commutator | C22×Q8 | C22×C4 |
| Frattini, cosocle | #391 | C23 |
| prime radical | C23 | C2×#99 |
| LCS | G ▷ C22×Q8 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C42 ◁ C4×C4⋊Q8 ◁ G | |
| LECS | G ▷ #391 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
323 conjugacy classes of subgroups; only the 56 normal subgroups are shown.
This group has 323 conjugacy classes of subgroups — too many to draw. Only its 56 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) = 64 |
A minimal faithful permutation representation — 3 generators on 48 points:
(1,2,5,9,6,10,4,8)(3,7,11,14,12,15,13,16)(17,18)(19,27,31,45,29,43,32,46)(20,33)(21,37)(22,24)(23,30,39,47,41,34,40,48)(25,35)(26,28)(36,44)(38,42);(1,3)(2,7)(4,13)(5,11)(6,12)(8,16)(9,14)(10,15)(17,19,28,32,22,29,21,31)(18,23,37,40,24,41,26,39)(20,34,38,47,36,30,35,48)(25,43,44,45,42,27,33,46);(1,4,6,5)(2,8,10,9)(3,11,12,13)(7,14,15,16)(18,24)(19,29)(25,42)(26,37)(30,34)(31,32)(33,44)(47,48)Polycyclic presentation:
2^9|a^2=e,b^2=fh,c^2=ghi,d^2=i,e^2=g,f^2=i,b^a=bd,c^a=ci,d^a=df,f^a=fh,c^b=cgh,d^b=dhi,e^b=ef,f^b=fi,g^b=gi,h^b=hi,e^d=ehi,f^d=fi41 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 57237);