Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2805613 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2112 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 36 points:
(2,4)(5,6)(7,12)(8,9)(10,11)(13,14)(15,19)(16,17)(18,20)(21,22,26,29)(23,24,31,27)(25,33,36,28)(30,35,34,32);(6,11)(9,12)(14,17)(18,19)(21,23,26,31)(22,27,29,24)(25,34,36,30)(28,32,33,35);(5,7,10,8)(6,12,11,9)(13,14)(15,18)(16,17)(19,20)(21,23)(22,27)(24,29)(25,30)(26,31)(28,35)(32,33)(34,36);(1,2)(3,4)(5,8,10,7)(6,12,11,9)(13,15,16,20)(14,18,17,19)(21,24,26,27)(22,23,29,31)(25,35,36,32)(28,34,33,30);(5,9,10,12)(6,8,11,7)(13,14)(15,19)(16,17)(18,20)(21,25,26,36)(22,28,29,33)(23,30,31,34)(24,32,27,35)Polycyclic presentation:
2^9|a^2=g,b^2=g,c^2=fh,d^2=f,e^2=fgh,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dg,e^b=ef,e^d=ef80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2720857);