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 | |
735 conjugacy classes of subgroups; only the 138 normal subgroups are shown.
This group has 735 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)(17,18,23,28)(19,29,32,42)(20,27,22,25)(21,26,35,41)(24,36,38,30)(31,43,46,47)(33,40,34,39)(37,45,48,44);(1,3,5,11)(2,6,8,15)(4,13,14,10)(7,16,12,9)(17,19)(18,24)(20,30)(21,31)(22,36)(23,32)(25,29)(26,37)(27,42)(28,38)(33,44)(34,45)(35,46)(39,43)(40,47)(41,48);(17,20)(18,25,28,27)(19,30)(21,34)(22,23)(24,29,38,42)(26,40,41,39)(31,45)(32,36)(33,35)(37,47,48,43)(44,46);(17,21,23,35)(18,26,28,41)(19,31,32,46)(20,33,22,34)(24,37,38,48)(25,39,27,40)(29,43,42,47)(30,44,36,45)Polycyclic presentation:
2^9|a^2=g,b^2=gi,c^2=hi,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, 391938);