Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #487707 |
| 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×C4⋊C4 | C24 |
| prime radical | #18218 | C2 |
| LCS | G ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×Q8 ◁ G | |
| LECS | G ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
857 conjugacy classes of subgroups; only the 118 normal subgroups are shown.
This group has 857 conjugacy classes of subgroups — too many to draw. Only its 118 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 — 4 generators on 40 points:
(2,6)(3,4)(7,8)(9,10)(11,16)(12,23)(13,18)(14,28)(15,20)(17,25)(19,29)(21,30)(22,32)(24,33)(26,39)(27,36)(31,38)(34,37)(35,40);(1,2)(3,6)(4,8)(5,7)(9,11,15,12)(10,16,20,23)(13,24,27,21)(14,25,29,22)(17,28,32,19)(18,33,36,30)(26,37,40,38)(31,35,34,39);(3,4)(6,8)(9,12,15,11)(10,17,20,32)(13,21,27,24)(14,22,29,25)(16,28,23,19)(18,34,36,31)(26,38,40,37)(30,35,33,39);(1,3,5,4)(2,6,7,8)(9,13,15,27)(10,18,20,36)(11,21,12,24)(14,26,29,40)(16,30,23,33)(17,31,32,34)(19,35,28,39)(22,37,25,38)Polycyclic presentation:
2^9|b^2=i,c^2=i,d^2=gi,e^2=g,f^2=i,b^a=be,c^a=cf,d^a=dg,e^a=eg,f^a=fi,h^a=hi,d^b=di,e^b=eg,f^b=fhi,d^c=dgi,e^c=eh,f^c=fi,f^e=fi44 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 395150);