Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33830 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D8 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| prime radical | C22×#508 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C22⋊C4 ◁ C8×C22⋊C4 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
1008 conjugacy classes of subgroups; only the 234 normal subgroups are shown.
This group has 1008 conjugacy classes of subgroups — too many to draw. Only its 234 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) = 256 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2)(3,4)(5,6,7,8)(10,16)(11,15)(12,20)(13,25)(17,33)(18,35)(19,32)(21,28)(22,37)(23,36)(24,30)(26,29)(31,38)(34,39);(9,10,14,19)(11,17,15,29)(12,22,24,35)(13,26,28,33)(16,21,32,25)(18,36,37,23)(20,39,30,31)(27,34,40,38);(2,4)(9,11,14,15)(10,17,19,29)(12,20,24,30)(13,21,28,25)(16,26,32,33)(18,34,37,38)(22,39,35,31)(23,27,36,40);(9,12,14,24)(10,18,19,37)(11,20,15,30)(13,27,28,40)(16,31,32,39)(17,34,29,38)(21,36,25,23)(22,33,35,26)Polycyclic presentation:
2^9|a^2=g,b^2=i,c^2=i,d^2=i,e^2=hi,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=eh,h^a=hi,d^b=de,e^b=eh,h^b=hi,e^d=eh,h^d=hi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 404633);