Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #13595 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#354 |
| commutator | C42 | C22×C8 |
| Frattini, cosocle | #384 | C23 |
| prime radical | C24 | #99 |
| LCS | G ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C23×C4 ◁ #2285 ◁ G | |
| LECS | G ▷ #384 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
316 conjugacy classes of subgroups; only the 82 normal subgroups are shown.
This group has 316 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:
(1,2,3,5,4,6,7,8)(9,10)(11,20,23,30)(12,16)(13,18,27,17)(14,19)(15,22,29,38)(21,35,37,31)(24,39,40,28)(25,33,36,32)(26,34);(9,11,13,23,14,22,27,38)(10,15,18,20,19,30,17,29)(12,24,25,40,26,21,36,37)(16,31,33,39,34,28,32,35);(9,12,14,26)(10,16,19,34)(11,21,22,24)(13,25,27,36)(15,28,30,31)(17,32,18,33)(20,35,29,39)(23,37,38,40)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=i,d^2=hi,e^2=g,f^2=i,b^a=bd,d^a=df,f^a=fh,h^a=hi,c^b=ci,d^b=di,e^b=efh,e^d=ehi,f^e=fi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72293125, #72293188, #72293211 |
| maximal normal subgroup of | #72293125, #72293188, #72293211 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 56812);