Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #150785 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22×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 | |
246 conjugacy classes of subgroups; only the 82 normal subgroups are shown.
This group has 246 conjugacy classes of subgroups — too many to draw. Only its 82 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) = 128 |
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)(19,23)(20,32)(21,35)(22,27)(24,28)(25,29)(26,33)(30,42)(31,38)(34,48)(36,44)(37,43)(39,47)(40,45)(41,46);(1,3)(2,6)(4,12)(5,11)(7,15)(8,9)(10,13)(14,16)(18,23)(19,28)(20,29)(21,33)(24,25)(26,42)(27,38)(30,47)(31,45)(32,36)(34,46)(35,37)(39,48)(40,41)(43,44);(1,4,5,13)(2,7,8,16)(3,10,11,12)(6,14,9,15)(17,19,29,44,21,30,34,45,22,31,46,47,33,43,20,28)(18,24,32,37,26,39,41,38,27,40,48,42,35,36,25,23)Polycyclic presentation:
2^9|a^2=e,c^2=fh,d^2=gh,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=cf,d^b=dg,e^b=efgh,f^b=fh,g^b=gh,h^b=hi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 57981);