Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #484601 |
| 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 | C2×#537 | C22 |
| LCS | G ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C22×C4⋊C4 ◁ G | |
| LECS | G ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
831 conjugacy classes of subgroups; only the 138 normal subgroups are shown.
This group has 831 conjugacy classes of subgroups — too many to draw. Only its 138 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:
(2,5)(3,7)(6,8)(9,10)(11,15)(12,22)(13,25)(14,18)(16,24)(17,26)(19,32)(20,28)(21,31)(23,37)(27,34)(29,35)(30,38)(33,36)(39,40);(1,2,4,6)(3,8,7,5)(9,11)(10,15)(12,19)(13,21)(14,20)(16,27)(17,31)(18,28)(22,36)(23,33)(24,34)(25,29)(26,35)(30,39)(32,37)(38,40);(9,12)(10,16,18,30)(11,19)(13,24)(14,23)(15,27,28,39)(17,22,25,37)(20,33)(21,34)(26,38)(29,32,31,36)(35,40);(11,20)(15,28)(19,33)(21,35)(27,39)(29,31)(32,36)(34,40)Polycyclic presentation:
2^9|b^2=g,c^2=hi,e^2=g,f^2=i,b^a=be,c^a=cf,e^a=eg,f^a=fi,h^a=hi,d^b=di,e^b=eg,f^b=fhi,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, 391914);