Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #454382 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C2×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C25 | C2×D4 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1046 conjugacy classes of subgroups; only the 226 normal subgroups are shown.
This group has 1046 conjugacy classes of subgroups — too many to draw. Only its 226 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:
(1,2,4,6)(3,7,8,5)(10,13)(11,15)(17,18)(19,21)(20,22)(23,24)(25,26,29,32)(27,30,34,38)(28,31,36,40)(33,37,35,39);(9,10,12,13)(11,14,15,16)(17,19,20,23)(18,21,22,24)(25,27,28,35,29,34,36,33)(26,30,31,39,32,38,40,37);(1,3,4,8)(2,5,6,7)(9,11,12,15)(10,14,13,16)(27,33)(28,36)(30,37)(31,40)(34,35)(38,39);(11,15)(14,16)(18,22)(21,24)(27,34)(30,38)(33,35)(37,39)Polycyclic presentation:
2^9|a^2=efi,b^2=egh,c^2=f,g^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=di,d^c=de,g^c=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 379736);