Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #93229 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C1 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #353 |
| commutator | C2×#97 | C2×C4 |
| Frattini, cosocle | #1985 | C22 |
| socle | C22 | #1352 |
| prime radical | C23 | #328 |
| LCS | G ▷ C2×#97 ▷ C2×C42 ▷ C23 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C4 ◁ C2×C42 ◁ #1985 ◁ G | |
| LECS | G ▷ #1985 ▷ C2×C42 ▷ C23 ▷ C22 ▷ C1 | |
123 conjugacy classes of subgroups; only the 26 normal subgroups are shown.
This group has 123 conjugacy classes of subgroups — too many to draw. Only its 26 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 — 2 generators on 48 points:
(1,2,4,9,6,11,3,12)(5,10,13,7,14,8,15,16)(17,18,21,25,22,27,29,26)(19,28,31,45,33,46,32,41)(20,34,36,42,38,44,37,24)(23,35,39,47,40,48,43,30);(1,3,5,13,6,4,14,15)(2,7,8,12,11,16,10,9)(17,19,29,31,22,33,21,32)(18,23,25,39,27,40,26,43)(20,35,36,47,38,48,37,30)(24,41,44,28,42,45,34,46)Polycyclic presentation:
2^9|a^2=d,b^2=egh,c^2=fghi,d^2=hi,e^2=hi,f^2=i,b^a=bc,c^a=ce,e^a=eg,g^a=gh,c^b=cf,d^b=defgi,e^b=ei,g^b=gi,d^c=dghi,e^c=ei,e^d=eh44 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 444);