Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #138564 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22×C4 |
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 | #19225 | 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 | |
313 conjugacy classes of subgroups; only the 80 normal subgroups are shown.
This group has 313 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) = 128 |
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)(18,24,28,33)(19,30)(20,25)(21,22)(26,41,43,48)(27,39,29,38)(31,45)(32,36)(34,35)(37,44,46,42);(1,3)(2,6,8,16)(4,12)(5,10)(7,14,15,9)(11,13)(17,18,25,38,21,27,20,33,23,28,34,39,22,29,35,24)(19,26,40,48,36,44,31,46,32,43,47,41,30,42,45,37);(1,3)(2,7)(4,12)(5,10)(6,14)(8,15)(9,16)(11,13)(17,19)(18,26)(20,31)(21,36)(22,30)(23,32)(24,37)(25,40)(27,44)(28,43)(29,42)(33,46)(34,47)(35,45)(38,48)(39,41)Polycyclic presentation:
2^9|a^2=f,b^2=dfg,d^2=ghi,e^2=h,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, 45760);