Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2076493 |
| automorphism | ? |
| 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 | |
2262 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)(5,6,9,11)(7,12,8,10)(13,14,17,19)(15,20,18,16)(21,22)(23,29)(24,28)(25,26)(27,30)(31,35)(32,33)(34,36);(5,7,9,8)(6,10,11,12)(13,15,17,18)(14,16,19,20)(22,26)(23,30)(28,32)(31,36);(6,11)(10,12)(13,16,17,20)(14,18,19,15)(21,22)(23,27)(24,32)(25,26)(28,33)(29,30)(31,35)(34,36);(1,2)(3,4)(5,8,9,7)(6,12,11,10)(21,23)(22,27)(24,31)(25,30)(26,29)(28,34)(32,35)(33,36);(5,6)(7,10)(8,12)(9,11)(13,14)(15,16)(17,19)(18,20)(21,24)(22,28)(23,31)(25,33)(26,32)(27,34)(29,35)(30,36)Polycyclic presentation:
2^9|a^2=hi,b^2=g,c^2=fgh,d^2=fh,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2027957);