Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3198 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C8 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator | C2×C4 | C23×C8 |
| Frattini, cosocle | C2×C42 | C24 |
| socle, prime radical | C23 | C2×C4×D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ #1744 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
534 conjugacy classes of subgroups; only the 234 normal subgroups are shown.
This group has 534 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) = 512 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2,4,7,5,3,6,8)(9,10)(11,18)(12,16)(13,22)(14,17)(15,20)(19,24)(21,23)(25,26)(27,34)(28,32)(29,38)(30,33)(31,36)(35,39)(37,40);(1,3,4,8,5,2,6,7)(9,11)(10,15)(12,21)(13,19)(14,20)(16,24)(17,18)(22,23)(25,26)(27,31)(28,38)(29,32)(30,33)(34,36)(35,40)(37,39);(9,12,14,13)(10,16,17,22)(11,19,20,21)(15,23,18,24)(25,27,29,36,30,37,28,35)(26,31,32,34,33,39,38,40);(9,13,14,12)(10,16,17,22)(11,19,20,21)(15,24,18,23)(25,28,30,29)(26,32,33,38)(27,35,37,36)(31,34,39,40)Polycyclic presentation:
2^9|a^2=g,b^2=g,c^2=ef,d^2=eh,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,c^b=ce,d^b=dh104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 126461);