Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #93784 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1347 |
| commutator | C22×C16 | C2×C4 |
| Frattini, cosocle | C2×C22⋊C16 | C22 |
| prime radical | #13304 | C2 |
| LCS | G ▷ C22×C16 ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C23×C8 ◁ C2×C22⋊C16 ◁ G | |
| LECS | G ▷ C2×C22⋊C16 ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
216 conjugacy classes of subgroups; only the 32 normal subgroups are shown.
This group has 216 conjugacy classes of subgroups — too many to draw. Only its 32 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 2 generators on 40 points:
(1,2)(3,7,8,5)(4,6)(9,10,12,19,16,22,15,21)(11,17,25,32,28,35,27,34)(13,30,24,20,14,31,29,39)(18,36,33,26,38,40,37,23);(2,5)(6,7)(10,17)(11,23)(12,15)(13,29)(14,24)(18,20)(19,32)(21,34)(22,35)(25,40)(26,28)(27,36)(30,37)(31,33)(38,39)Polycyclic presentation:
2^9|a^2=d,c^2=fh,d^2=i,e^2=hi,f^2=hi,g^2=i,h^2=i,b^a=bc,c^a=ce,e^a=eg,f^a=fh,g^a=gi,h^a=hi,c^b=cf,d^b=defh,e^b=eh,f^b=fh,g^b=gi,h^b=hi,d^c=dghi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 982);