Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #484162 |
| 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 | #3449 | C22 |
| LCS | G ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
830 conjugacy classes of subgroups; only the 130 normal subgroups are shown.
This group has 830 conjugacy classes of subgroups — too many to draw. Only its 130 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) = 128 |
A minimal faithful permutation representation — 4 generators on 40 points:
(2,6)(3,4)(7,8)(9,10)(11,19)(12,16)(13,26)(14,18)(15,21)(17,28)(20,35)(22,34)(23,37)(24,32)(25,40)(27,30)(29,36)(31,38)(33,39);(1,2)(3,6)(4,8)(5,7)(9,11,14,22)(10,15,18,30)(12,23,24,20)(13,27,28,21)(16,31,32,29)(17,34,26,19)(25,36,39,38)(33,35,40,37);(1,3,5,4)(2,6,7,8)(12,24)(13,28)(16,32)(20,23)(21,27)(29,31)(33,40)(35,37);(9,12,14,24)(10,16,18,32)(11,20,22,23)(13,25,28,39)(15,29,30,31)(17,33,26,40)(19,35,34,37)(21,36,27,38)Polycyclic presentation:
2^9|b^2=i,c^2=gi,d^2=i,e^2=g,b^a=be,c^a=cf,e^a=eg,h^a=hi,d^b=di,e^b=eg,f^b=fhi,d^c=di,e^c=eh,f^e=fi50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 391572);