Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24398 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#121 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | C2×#389 |
| prime radical | C25 | C2×Q8 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1280 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)(7,11)(8,19,22,12)(9,14)(10,23)(13,15)(16,25,32,36)(17,30)(18,26)(20,33,35,28)(21,29,31,34)(24,27);(2,4)(5,7,15,17,9,18,10,24)(6,11,13,30,14,26,23,27)(8,20,31,32,22,35,21,16)(12,28,29,25,19,33,34,36);(5,8,10,21,9,22,15,31)(6,12,13,29,14,19,23,34)(7,16,17,20,18,32,24,35)(11,25,27,28,26,36,30,33);(2,4)Polycyclic presentation:
2^9|a^2=h,b^2=fgh,c^2=g,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,g^a=gi,c^b=cg,f^b=fi,g^b=gi,f^c=fi,h^c=hi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 277297);