Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24292 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | D42 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #2361 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
899 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)(5,6,10,15,11,16,9,14)(7,17,20,29,21,28,19,12)(8,22,26,31,24,32,25,13)(18,34,36,30,35,27,23,33);(5,7,9,19,11,21,10,20)(6,12,15,17,16,29,14,28)(8,23,25,35,24,36,26,18)(13,30,22,27,31,33,32,34);(2,4)(5,8,9,25,11,24,10,26)(6,13,15,22,16,31,14,32)(7,18,19,23,21,35,20,36)(12,27,17,33,29,34,28,30);(6,14,16,15)(8,24)(12,28,29,17)(13,22,31,32)(18,35)(23,36)(25,26)(27,33,34,30)Polycyclic presentation:
2^9|a^2=h,b^2=e,c^2=e,d^2=fh,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,c^b=cg,g^b=gi,d^c=di,g^c=gi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 284875);