Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #130730 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1786 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | #61084 | C2 |
| LCS | G ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ #2931 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
810 conjugacy classes of subgroups; only the 98 normal subgroups are shown.
This group has 810 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) = 128 |
A minimal faithful permutation representation — 3 generators on 40 points:
(1,2,3,6,4,5,7,8)(9,10,14,21,15,22,12,19,16,23,29,39,13,20,30,34)(11,24,27,17,26,36,32,35,28,40,33,37,31,18,25,38);(2,5)(3,7)(9,11)(10,17)(12,27)(13,26)(14,32)(15,31)(16,28)(18,34)(19,36)(20,35)(21,40)(22,38)(23,37)(24,39)(25,29)(30,33);(2,5)(3,7)(9,11)(10,18)(12,25)(13,31)(14,33)(15,26)(16,28)(17,34)(19,37)(20,40)(21,35)(22,24)(23,36)(27,29)(30,32)(38,39)Polycyclic presentation:
2^9|a^2=f,d^2=hi,e^2=i,f^2=gh,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, 38244);