Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #484587 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×B(4,𝔽2) |
| commutator, Frattini, cosocle | C2×C22⋊C4 | C24 |
| prime radical | C24 | C22×D4 |
| LCS | G ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
741 conjugacy classes of subgroups; only the 138 normal subgroups are shown.
This group has 741 conjugacy classes of subgroups — too many to draw. Only its 138 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 — 4 generators on 48 points:
(1,2,5,8)(3,9,11,16)(4,12,14,7)(6,13,15,10)(17,18)(19,28)(20,26)(21,25)(22,27)(23,29)(24,33)(30,42)(31,36)(32,40)(34,39)(35,41)(37,43)(38,44)(45,48)(46,47);(1,3,5,11)(2,6,8,15)(4,13,14,10)(7,16,12,9)(17,19,22,31)(18,23,27,38)(20,29,33,44)(21,30,35,46)(24,36,26,28)(25,37,41,48)(32,43,34,45)(39,47,40,42);(17,20,22,33)(18,24)(19,29,31,44)(21,34,35,32)(23,36)(25,40)(26,27)(28,38)(30,45,46,43)(37,42)(39,41)(47,48);(17,21,22,35)(18,25,27,41)(19,30,31,46)(20,32,33,34)(23,37,38,48)(24,39,26,40)(28,42,36,47)(29,43,44,45)Polycyclic presentation:
2^9|a^2=g,b^2=gi,c^2=h,d^2=i,e^2=g,b^a=be,c^a=cf,e^a=eg,h^a=hi,e^b=eg,f^b=fhi,d^c=di,e^c=eh,f^c=fi,f^e=fi50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 391936);