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 | |
1890 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 36 points:
(6,11)(8,12)(15,19)(18,20)(21,22,25,28)(23,26,29,33)(24,31,32,27)(30,36,35,34);(2,4)(5,6,10,11)(7,12,9,8)(14,17)(18,20)(23,29)(26,33)(30,35)(34,36);(1,2,3,4)(5,7,10,9)(6,12,11,8)(13,14)(15,18)(16,17)(19,20)(21,22)(23,26)(24,27)(25,28)(29,33)(30,34)(31,32)(35,36);(5,8)(6,7)(9,11)(10,12)(13,15,16,19)(14,18,17,20)(21,23)(22,26)(24,30)(25,29)(27,34)(28,33)(31,36)(32,35);(5,9,10,7)(6,12,11,8)(21,24,25,32)(22,27,28,31)(23,30,29,35)(26,34,33,36)Polycyclic presentation:
2^9|a^2=g,b^2=f,c^2=hi,d^2=fi,e^2=fg,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, 2849443);