Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30490 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | D42 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #2366 |
| prime radical | C24 | H(5,𝔽2) |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
722 conjugacy classes of subgroups; only the 226 normal subgroups are shown.
This group has 722 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) = 64 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2)(3,5)(4,6)(7,8)(9,10,15,20,16,22,14,21)(11,23,26,17,27,34,32,35)(12,18,29,37,30,19,13,31)(24,38,28,33,39,36,25,40);(1,3,4,7)(2,5,6,8)(9,11)(10,17)(12,28)(13,24)(14,32)(15,26)(16,27)(18,36)(19,38)(20,34)(21,23)(22,35)(25,30)(29,39)(31,33)(37,40);(2,6)(5,8)(9,12,14,13,16,30,15,29)(10,18,21,31,22,19,20,37)(11,24,26,28,27,39,32,25)(17,33,34,36,35,40,23,38);(2,6)(5,8)(9,13,14,30,16,29,15,12)(10,19,20,31,22,18,21,37)(11,25,26,24,27,28,32,39)(17,33,23,38,35,40,34,36)Polycyclic presentation:
2^9|a^2=h,b^2=f,c^2=g,d^2=e,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,c^b=cg,d^b=de,e^b=ei,g^b=gi,d^c=di68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 293483);