Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #454039 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2410 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #60369 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2198 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2)(3,4)(6,11)(8,19)(9,22)(12,28)(14,23)(15,26)(16,20)(17,24)(21,25)(27,31)(30,36)(34,35);(5,6)(7,15)(8,13)(9,23)(10,14)(11,22)(12,18)(16,33)(17,28)(19,27)(20,30)(21,32)(24,26)(25,31)(29,34)(35,36);(5,7,10,18)(6,12,14,15)(8,20,21,35)(9,24,22,17)(11,26,23,28)(13,30,32,36)(16,25,34,19)(27,33,31,29);(2,4)(5,8)(6,13)(7,16)(9,25)(10,21)(11,27)(12,29)(14,32)(15,33)(17,35)(18,34)(19,22)(20,24)(23,31)(26,30)(28,36)Polycyclic presentation:
2^9|c^2=i,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,e^b=ei,f^b=fi,h^b=hi,d^c=de,e^c=ei,h^c=hi,e^d=ei,g^d=gi,h^d=hi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 361286);