Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #476144 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2675 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C24 | 2−1+4 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
654 conjugacy classes of subgroups; only the 178 normal subgroups are shown.
This group has 654 conjugacy classes of subgroups — too many to draw. Only its 178 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 — 4 generators on 40 points:
(1,2,5,8)(3,6,7,4)(9,10)(11,20)(12,25)(13,17)(14,30)(15,19)(16,23)(18,31)(21,32)(22,37)(24,38)(26,33)(27,36)(28,40)(29,35)(34,39);(1,3)(2,6)(4,8)(5,7)(9,11,14,23,15,24,31,39)(10,16,18,20,19,34,30,38)(12,26,13,28,27,21,29,22)(17,32,25,40,35,33,36,37);(1,4,5,6)(2,7,8,3)(9,12,15,27)(10,17,19,35)(11,21,24,26)(13,14,29,31)(16,32,34,33)(18,36,30,25)(20,37,38,40)(22,23,28,39);(2,8)(3,7)(9,13,14,27,15,29,31,12)(10,17,18,25,19,35,30,36)(11,22,23,26,24,28,39,21)(16,33,20,37,34,32,38,40)Polycyclic presentation:
2^9|a^2=efi,b^2=e,c^2=ef,d^2=e,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,d^b=def,g^b=gi,d^c=de,e^c=ei,f^c=fi,h^c=hi,h^d=hi50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 384678);