Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #418488 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2688 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
959 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)(5,6,11,17)(7,18,22,34)(8,23,27,33)(9,16,10,15)(12,28,31,21)(13,25,14,26)(19,29,24,32)(20,30,36,35);(5,7,9,21,11,22,10,28)(6,12,15,18,17,31,16,34)(8,24,26,20,27,19,25,36)(13,32,23,35,14,29,33,30);(5,8)(6,13)(7,19)(9,25)(10,26)(11,27)(12,29)(14,17)(15,33)(16,23)(18,35)(20,21)(22,24)(28,36)(30,34)(31,32);(2,4)(5,8)(6,14)(7,20,22,36)(9,25)(10,26)(11,27)(12,30,31,35)(13,17)(15,23)(16,33)(18,29,34,32)(19,21,24,28)Polycyclic presentation:
2^9|a^2=i,b^2=f,d^2=g,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cg,d^b=de,g^b=gi,h^b=hi,f^c=fi,h^c=hi,f^d=fi,h^d=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 355849);