Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23918 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| socle, prime radical | C23 | C2×C4×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C42×C8 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
646 conjugacy classes of subgroups; only the 210 normal subgroups are shown.
This group has 646 conjugacy classes of subgroups — too many to draw. Only its 210 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)(3,6)(4,7)(5,8)(9,10,13,16)(11,17,20,24)(12,21,18,15)(14,22,23,19)(25,26,29,33,30,27,31,35)(28,32,37,39,36,34,38,40);(1,3,6,2)(4,7,5,8)(25,27,29,35,30,26,31,33)(28,34,37,40,36,32,38,39);(1,4,6,5)(2,7,3,8)(9,11,18,22,13,20,12,19)(10,14,21,24,16,23,15,17)(25,28)(26,32)(27,34)(29,37)(30,36)(31,38)(33,39)(35,40);(1,5,6,4)(2,7,3,8)(28,36)(32,34)(37,38)(39,40)Polycyclic presentation:
2^9|a^2=g,b^2=egh,c^2=efh,d^2=e,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,c^b=ce,d^c=di104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 126564);