Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #1506685 |
| 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 | |
2422 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 36 points:
(6,7)(8,12)(15,16)(18,19)(21,22,26,27)(23,28,29,24)(25,30,33,34)(31,35,36,32);(2,4)(6,7)(10,11)(13,14,17,20)(15,18,16,19)(22,27)(23,29)(25,33)(32,35);(1,2)(3,4)(5,6,9,7)(8,10,12,11)(13,15,17,16)(14,18,20,19)(21,23)(22,28)(24,27)(25,31)(26,29)(30,35)(32,34)(33,36);(5,7,9,6)(8,11,12,10)(13,14)(15,18)(16,19)(17,20)(21,24,26,28)(22,29,27,23)(25,32,33,35)(30,36,34,31);(5,8,9,12)(6,10,7,11)(13,16,17,15)(14,19,20,18)(21,25)(22,30)(23,31)(24,32)(26,33)(27,34)(28,35)(29,36)Polycyclic presentation:
2^9|a^2=f,b^2=fgi,c^2=g,d^2=i,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dg,e^b=ef,d^c=df74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1457290);