Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #130730 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1786 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C23×C8 | C23 |
| prime radical | #61049 | C2 |
| LCS | G ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ #2929 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
904 conjugacy classes of subgroups; only the 98 normal subgroups are shown.
This group has 904 conjugacy classes of subgroups — too many to draw. Only its 98 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 — 3 generators on 40 points:
(1,2)(3,7,8,5)(4,6)(9,10,14,21,16,20,12,28,15,22,31,40,13,32,30,19)(11,23,26,17,25,18,24,37,27,39,29,36,33,38,34,35);(1,3)(2,5)(4,8)(6,7)(9,11)(10,17)(12,29)(13,25)(14,34)(15,27)(16,33)(18,19)(20,35)(21,23)(22,36)(24,31)(26,30)(28,38)(32,37)(39,40);(1,3)(2,5)(4,8)(6,7)(9,11)(10,18)(12,24)(13,33)(14,26)(15,27)(16,25)(17,19)(20,39)(21,37)(22,38)(23,32)(28,36)(29,31)(30,34)(35,40)Polycyclic presentation:
2^9|a^2=f,d^2=hi,e^2=i,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eg,d^b=dh,f^b=fgh,h^b=hi,e^c=ei,f^c=fgi53 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 37926);