Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5211343 |
| 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 | C22×#121 | C22 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2334 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)(7,9)(8,10)(11,12)(13,14)(15,19)(16,17)(18,20)(21,22,26,29)(23,28,32,27)(24,33,35,25)(30,36,34,31);(8,12)(10,11)(13,14)(15,19)(16,17)(18,20)(21,23,26,32)(22,27,29,28)(24,34,35,30)(25,31,33,36);(6,7)(10,11)(14,17)(18,19)(21,23)(22,28)(24,34)(25,36)(26,32)(27,29)(30,35)(31,33);(1,2,3,4)(5,7,9,6)(8,11,12,10)(13,14)(15,18)(16,17)(19,20)(21,24)(22,25)(23,30)(26,35)(27,36)(28,31)(29,33)(32,34);(5,8)(6,10)(7,11)(9,12)(13,15)(14,18)(16,20)(17,19)(21,25)(22,24)(23,31)(26,33)(27,34)(28,30)(29,35)(32,36)Polycyclic presentation:
2^9|a^2=f,b^2=f,d^2=fi,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^b=df,e^b=eg,d^c=dfg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5184358);