Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #60372 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C23×D4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| socle | C22 | C23×C4⋊C4 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2561 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 36 points:
(2,4)(5,6,10,14,11,15,9,23)(7,16,18,12,19,26,24,27)(8,13,22,29,20,28,21,33)(17,30,32,25,31,36,34,35);(5,7,10,18,11,19,9,24)(6,12,14,26,15,27,23,16)(8,17,22,32,20,31,21,34)(13,25,29,36,28,35,33,30);(1,2,3,4);(5,8)(6,13)(7,17)(9,21)(10,22)(11,20)(12,25)(14,29)(15,28)(16,30)(18,32)(19,31)(23,33)(24,34)(26,36)(27,35);(8,20)(13,28)(17,31)(21,22)(25,35)(29,33)(30,36)(32,34)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=g,f^2=i,h^2=i,b^a=bf,c^a=cg,f^a=fi,f^b=fi,e^d=ei122 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6279117);