Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12155 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | B(4,𝔽2) |
| commutator, Frattini, cosocle | C2×C4⋊C4 | C24 |
| socle | C22 | C2×B(4,𝔽2) |
| prime radical | #3449 | C22 |
| LCS | G ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C4 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
822 conjugacy classes of subgroups; only the 138 normal subgroups are shown.
This group has 822 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)(12,17)(13,16)(14,19)(18,26)(20,29)(21,28)(22,30)(23,31)(24,33)(25,27)(32,37)(34,39)(35,40)(36,38);(1,2,4,6)(3,8,7,5)(9,11)(10,15)(12,21)(13,25)(14,20)(16,30)(17,28)(18,27)(19,29)(22,26)(23,36)(24,34)(31,40)(32,38)(33,39)(35,37);(10,16,19,18)(11,20)(15,27,29,30)(17,31,33,32)(21,34)(22,25)(28,38,39,40)(35,36);(9,12,14,24)(10,17,19,33)(11,21,20,34)(13,23,26,37)(15,28,29,39)(16,31,18,32)(22,35,25,36)(27,38,30,40)Polycyclic presentation:
2^9|b^2=g,c^2=h,d^2=i,e^2=g,f^2=i,b^a=be,c^a=cf,e^a=eg,f^a=fi,h^a=hi,c^b=ci,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, 391912);