Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #232138 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2380 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C24 | H(5,𝔽2) |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
680 conjugacy classes of subgroups; only the 226 normal subgroups are shown.
This group has 680 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:
(2,5)(3,7)(9,10)(11,22)(12,18,29,17)(13,30,31,37)(14,21)(15,20)(16,25)(19,28)(23,36,39,38)(24,33,27,40)(26,35)(32,34);(1,2,4,5)(3,8,7,6)(9,11,15,26)(10,16,20,34)(12,27,29,24)(13,23,31,39)(14,32,28,25)(17,36,18,38)(19,35,21,22)(30,40,37,33);(9,12,28,31,15,29,14,13)(10,17,19,30,20,18,21,37)(11,23,32,24,26,39,25,27)(16,33,22,38,34,40,35,36);(1,3,4,7)(2,6,5,8)(9,13,14,29,15,31,28,12)(10,18,19,37,20,17,21,30)(11,24,25,23,26,27,32,39)(16,33,22,38,34,40,35,36)Polycyclic presentation:
2^9|a^2=h,b^2=efg,c^2=gi,d^2=e,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,f^a=fi,c^b=cg,d^b=de,e^b=ei,g^b=gi,h^c=hi,h^d=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 294394);