Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1655 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C8 | #120 |
| commutator | C42 | C22×C8 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | #1850 |
| prime radical | #121 | M16 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C8 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
253 conjugacy classes of subgroups; only the 88 normal subgroups are shown.
This group has 253 conjugacy classes of subgroups — too many to draw. Only its 88 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:
(1,2,7,12)(3,13,4,15)(5,11,6,10)(8,16,9,14)(17,18,23,28,24,19,29,32,21,27,37,44,22,31,25,39)(20,33,35,47,34,26,41,42,36,30,45,46,38,43,40,48);(1,3,7,4)(2,8,12,9)(5,16,6,14)(10,15,11,13)(17,19,23,32,24,27,29,44,21,31,37,39,22,18,25,28)(20,30,35,46,34,43,41,48,36,33,45,47,38,26,40,42);(1,4,7,3)(2,9,12,8)(5,16,6,14)(10,15,11,13)(17,20)(18,26)(19,30)(21,36)(22,38)(23,35)(24,34)(25,40)(27,43)(28,42)(29,41)(31,33)(32,46)(37,45)(39,47)(44,48)Polycyclic presentation:
2^9|a^2=f,b^2=f,c^2=g,d^2=g,e^2=gi,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=egi,d^b=dg,e^b=ei,d^c=di,e^c=egi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 43071);