Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #133908 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1782 |
| commutator | C23×C4 | C22×C4 |
| Frattini, cosocle | C2×#117 | C23 |
| prime radical | C23×D4 | C2×C4 |
| LCS | G ▷ C23×C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ #2296 ◁ G | |
| LECS | G ▷ C2×#117 ▷ C23 ▷ C2 ▷ C1 | |
730 conjugacy classes of subgroups; only the 82 normal subgroups are shown.
This group has 730 conjugacy classes of subgroups — too many to draw. Only its 82 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:
(1,2,3,6,4,5,7,8)(9,10)(11,19,22,35)(12,17)(13,25,27,16)(14,18)(15,20,30,36)(21,38)(23,33)(24,32,39,31)(26,34)(28,40)(29,37);(2,5)(3,7)(9,11)(10,15,18,30)(12,20)(13,26)(14,22)(16,28,25,29)(17,19,33,35)(21,27)(23,36)(24,40)(31,38,32,34)(37,39);(10,16)(12,23)(15,28)(17,32)(18,25)(19,34)(20,36)(24,39)(29,30)(31,33)(35,38)(37,40)Polycyclic presentation:
2^9|a^2=f,b^2=h,d^2=i,f^2=g,b^a=bd,c^a=ce,d^a=dg,e^a=eh,h^a=hi,e^b=ei,f^b=fgi,d^c=di,f^c=fh,f^e=fi44 conjugacy classes — view the full character table.
| maximal subgroup of | #72283599, #72283699, #72283735, … |
| maximal normal subgroup of | #72283599, #72283699, #72283735, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 41357);