Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24390 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C2×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C23 | C22×C4⋊C4 |
| prime radical | C24 | C22×Q8 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C2×#376 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
770 conjugacy classes of subgroups; only the 266 normal subgroups are shown.
This group has 770 conjugacy classes of subgroups — too many to draw. Only its 266 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 — 4 generators on 48 points:
(1,2)(3,7)(4,5)(6,8)(9,10)(11,15)(12,14)(13,16)(17,18,22,26,23,27,21,36)(19,24,31,39,29,38,30,45)(20,32,35,40,33,46,34,25)(28,41,43,47,44,48,42,37);(2,5)(6,7)(9,11,12,16)(10,13,14,15)(17,19)(18,24)(20,28)(21,30)(22,31)(23,29)(25,37)(26,39)(27,38)(32,41)(33,44)(34,42)(35,43)(36,45)(40,47)(46,48);(11,16)(13,15)(17,20,21,34,23,33,22,35)(18,25,26,32,27,40,36,46)(19,28,30,42,29,44,31,43)(24,37,39,41,38,47,45,48);(1,3,4,8)(2,6,5,7)(10,14)(13,15)(19,29)(20,33)(24,38)(25,40)(30,31)(32,46)(34,35)(39,45)Polycyclic presentation:
2^9|a^2=h,b^2=g,c^2=f,d^2=eg,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,c^b=cg,d^b=di,d^c=di92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 271504);