Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30484 |
| 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 | |
732 conjugacy classes of subgroups; only the 226 normal subgroups are shown.
This group has 732 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)(9,10,14,21,16,22,15,20)(11,23,27,36,28,40,26,17)(12,18,13,32,31,19,30,38)(24,37,29,35,33,39,25,34);(1,3,4,7)(2,5,6,8)(9,11)(10,17)(12,29)(13,33)(14,27)(15,26)(16,28)(18,37)(19,39)(20,40)(21,23)(22,36)(24,30)(25,31)(32,35)(34,38);(2,6)(5,8)(9,12,15,30,16,31,14,13)(10,18,20,38,22,19,21,32)(11,24,27,29,28,33,26,25)(17,34,23,37,36,35,40,39);(2,6)(5,8)(9,13,14,31,16,30,15,12)(10,19,20,32,22,18,21,38)(11,25,26,33,28,29,27,24)(17,35,23,39,36,34,40,37)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=gi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 293482);