Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23414 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#127 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | C2×#399 |
| prime radical | C23×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2634 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,4)(5,6,10,14,11,15,9,24)(7,16,20,26,19,31,18,12)(8,13,23,29,21,28,22,36)(17,32,35,25,33,30,34,27);(1,2,3,4)(5,7,9,18,11,19,10,20)(6,12,14,16,15,26,24,31)(8,17,22,34,21,33,23,35)(13,27,29,32,28,25,36,30);(5,8)(6,13)(7,17)(9,22)(10,23)(11,21)(12,25)(14,29)(15,28)(16,30)(18,34)(19,33)(20,35)(24,36)(26,27)(31,32);(8,21)(13,28)(17,33)(22,23)(25,27)(29,36)(30,32)(34,35)Polycyclic presentation:
2^9|a^2=h,b^2=e,e^2=i,h^2=i,b^a=be,c^a=cf,e^a=ei,g^a=gi,c^b=cg,f^b=fi,d^c=di92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 264879);