Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #150138 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#357 |
| commutator | C2×C16 | C22×C4 |
| Frattini, cosocle | C4×C16 | C23 |
| prime radical | C22×D16 | C4 |
| LCS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C2×C4×C8 ◁ C2×C4×C16 ◁ G | |
| LECS | G ▷ C4×C16 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
268 conjugacy classes of subgroups; only the 76 normal subgroups are shown.
This group has 268 conjugacy classes of subgroups — too many to draw. Only its 76 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)(9,10,11,19,14,21,15,17)(12,24,22,32,27,16,28,33)(13,29,23,38,31,36,30,20)(18,37,34,25,39,40,35,26);(10,16)(11,15)(12,25)(13,30)(17,33)(18,38)(19,32)(20,39)(21,24)(22,37)(23,31)(26,27)(28,40)(29,34)(35,36);(2,5)(6,8)(9,11,14,15)(10,17,21,19)(12,22,27,28)(13,23,31,30)(16,32,24,33)(18,35,39,34)(20,36,38,29)(25,40,26,37)Polycyclic presentation:
2^9|a^2=e,c^2=i,d^2=fhi,e^2=g,f^2=hi,h^2=i,b^a=bd,c^a=cg,d^a=df,f^a=fh,h^a=hi,c^b=ci,d^b=dfi,e^b=ei,f^b=fh,h^b=hi74 conjugacy classes — view the full character table.
| maximal subgroup of | #72232028, #72234322, #72234323, … |
| maximal normal subgroup of | #72232028, #72234322, #72234323, … |
| maximal quotient of | #72286707, C2×#150142, #72322645, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 57338);