Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #418438 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#121 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#525 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1714 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2)(3,4)(9,10,14,18)(11,19,22,34)(12,23,26,16)(13,27,29,17)(15,28,31,21)(20,32,35,36)(24,30,38,40)(25,33,39,37);(5,6)(7,8)(9,11)(10,15)(12,24)(13,28)(14,22)(16,32)(17,34)(18,31)(19,27)(20,25)(21,29)(23,36)(26,38)(30,33)(35,39)(37,40);(2,4)(6,8);(9,12,13,25,14,26,29,39)(10,16,17,33,18,23,27,37)(11,20,21,24,22,35,28,38)(15,30,19,32,31,40,34,36)Polycyclic presentation:
2^9|a^2=i,d^2=e,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,d^b=de,e^b=ei,h^b=hi,h^d=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 325636);