Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3290 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | D42 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| socle | C22 | C2×D42 |
| prime radical | C2×#3067 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2516 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 36 points:
(1,2,4,3)(5,6,10,14,11,8,22,15)(7,16,20,17,21,28,19,12)(9,13,25,30,26,23,33,31)(18,29,24,32,35,36,34,27);(1,3,4,2)(5,7,11,21)(6,12,8,17)(9,24,26,34)(10,19,22,20)(13,29,23,36)(14,28,15,16)(18,33,35,25)(27,31,32,30);(5,8)(6,11)(7,17)(9,23)(10,15)(12,21)(13,26)(14,22)(16,19)(18,32)(20,28)(24,36)(25,31)(27,35)(29,34)(30,33);(5,9,10,25,11,26,22,33)(6,13,14,30,8,23,15,31)(7,18,20,24,21,35,19,34)(12,27,16,29,17,32,28,36);(5,10,11,22)(6,14,8,15)(7,19,21,20)(9,25,26,33)(12,28,17,16)(13,30,23,31)(18,34,35,24)(27,36,32,29)Polycyclic presentation:
2^9|a^2=h,b^2=fh,d^2=f,e^2=i,f^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,g^a=gi,d^b=df,f^b=fi,h^b=hi,g^c=gi116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6289197);