Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #454369 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2669 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #23406 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1845 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)(5,6)(7,17)(8,18)(9,14)(10,28)(11,16)(12,19)(13,24)(15,30)(20,25)(21,27)(22,32)(23,33)(26,36)(29,35)(31,34);(5,7,10,19,11,20,30,32)(6,12,15,17,16,22,28,25)(8,21,23,36,24,14,29,34)(9,18,26,35,27,13,31,33);(2,4)(5,8)(6,13)(7,14)(9,25)(10,29)(11,24)(12,31)(15,35)(16,18)(17,27)(19,36)(20,21)(22,26)(23,30)(28,33)(32,34);(5,9,10,26,11,27,30,31)(6,14,15,34,16,21,28,36)(7,18,19,35,20,13,32,33)(8,22,23,25,24,12,29,17)Polycyclic presentation:
2^9|b^2=e,d^2=e,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,g^b=gi,d^c=de,e^c=ei,h^c=hi,h^d=hi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 361565);