Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #138564 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1893 |
| commutator | C4×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | #19227 | C2 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ #2966 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
306 conjugacy classes of subgroups; only the 80 normal subgroups are shown.
This group has 306 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:
(1,2,5,8)(3,6)(4,9,13,14)(7,12)(10,16)(11,15)(17,18,22,28,24,29,23,27)(19,30,33,31,34,40,26,21)(20,35,38,42,39,43,37,25)(32,36,46,44,47,45,41,48);(1,3)(2,6,8,16)(4,12)(5,10)(7,14,15,9)(11,13)(17,19,31,29,23,33,30,27,24,34,21,18,22,26,40,28)(20,32,45,42,37,46,44,43,39,47,36,25,38,41,48,35);(1,3)(2,7)(4,12)(5,10)(6,14)(8,15)(9,16)(11,13)(17,20)(18,25)(19,32)(21,36)(22,38)(23,37)(24,39)(26,41)(27,43)(28,35)(29,42)(30,44)(31,45)(33,46)(34,47)(40,48)Polycyclic presentation:
2^9|a^2=f,b^2=dfgi,d^2=ghi,e^2=h,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,g^a=gi,d^b=dh,e^b=ehi,f^b=fh,d^c=dhi,e^c=eh,f^c=fhi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 45763);