Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3057 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C8 | C22×D4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| socle | C22 | C22×C4×D4 |
| prime radical | #60643 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2984 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 36 points:
(2,4)(5,6,11,16,12,17,10,26)(7,18,9,15,21,31,20,13)(8,14,25,30,23,29,24,35)(19,32,22,28,33,36,34,27);(5,7,10,20,12,21,11,9)(6,13,16,18,17,15,26,31)(8,19,24,34,23,33,25,22)(14,27,30,32,29,28,35,36);(1,2,3,4)(5,8)(6,14)(7,19)(9,22)(10,24)(11,25)(12,23)(13,27)(15,28)(16,30)(17,29)(18,32)(20,34)(21,33)(26,35)(31,36);(5,9,11,21,12,20,10,7)(6,15,16,31,17,13,26,18)(8,22,25,33,23,34,24,19)(14,28,30,36,29,27,35,32);(8,23)(14,29)(19,33)(22,34)(24,25)(27,28)(30,35)(32,36)Polycyclic presentation:
2^9|a^2=h,b^2=f,c^2=g,d^2=h,f^2=i,h^2=i,b^a=bf,c^a=cg,f^a=fi,e^c=ei128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6280986);