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 | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2016 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 36 points:
(1,2,3,4)(5,6)(7,12)(8,9)(10,11)(13,14,18,17)(15,19,16,20)(22,23)(25,33)(27,29)(31,34);(6,11)(9,12)(13,14)(15,20)(16,19)(17,18)(21,22,26,23)(24,29,32,27)(25,30,33,28)(31,35,34,36);(5,7,10,8)(6,12,11,9)(13,15,18,16)(14,19,17,20)(21,23,26,22)(24,29,32,27)(25,30,33,28)(31,36,34,35);(1,2)(3,4)(5,8,10,7)(6,12,11,9)(13,16,18,15)(14,19,17,20)(21,24)(22,27)(23,29)(25,34)(26,32)(28,36)(30,35)(31,33);(5,9,10,12)(6,8,11,7)(13,17,18,14)(15,20,16,19)(21,25,26,33)(22,28,23,30)(24,31,32,34)(27,35,29,36)Polycyclic presentation:
2^9|a^2=fghi,b^2=g,c^2=f,d^2=fg,e^2=f,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, 2722604);