Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #485337 |
| 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 | #1595 | 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 | |
870 conjugacy classes of subgroups; only the 120 normal subgroups are shown.
This group has 870 conjugacy classes of subgroups — too many to draw. Only its 120 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:
(1,2)(3,5)(4,6)(7,8)(9,10)(11,21)(12,19)(13,26)(14,17)(15,20)(16,22)(18,28)(23,37)(24,34)(25,32)(27,33)(29,31)(30,38)(35,39)(36,40);(2,5)(3,8)(6,7)(9,11)(10,16)(12,22)(13,23)(14,29)(15,24)(17,21)(18,30)(19,34)(20,31)(25,36)(26,40)(27,38)(28,39)(32,35)(33,37);(9,12)(10,17,20,19)(11,22)(13,27)(14,15)(16,21,31,34)(18,33,26,32)(23,38)(24,29)(25,28)(30,37,40,35)(36,39);(2,6)(5,7)(9,13,15,28)(10,18,20,26)(11,23,24,39)(12,25,14,27)(16,30,31,40)(17,32,19,33)(21,35,34,37)(22,36,29,38)Polycyclic presentation:
2^9|c^2=hi,d^2=i,e^2=g,b^a=be,c^a=cf,d^a=dg,e^a=eg,h^a=hi,e^b=eg,f^b=fhi,d^c=di,e^c=eh,f^c=fi,f^e=fi41 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 392535);