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 | |
2503 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)(8,11)(10,12)(13,14,17,15)(16,19,20,18)(21,22,27,29)(23,30,24,26)(25,33,35,28)(31,34,32,36);(6,7)(8,12)(13,14)(15,17)(16,19)(18,20)(21,23,27,24)(22,26,29,30)(25,31,35,32)(28,34,33,36);(6,7)(10,11)(14,15)(18,19)(21,24,27,23)(22,26,29,30)(25,31,35,32)(28,36,33,34);(1,2,3,4)(5,7,9,6)(8,11,12,10)(13,15,17,14)(16,19,20,18)(21,25,27,35)(22,28,29,33)(23,31,24,32)(26,34,30,36);(5,8)(6,10)(7,11)(9,12)(13,16)(14,18)(15,19)(17,20)(21,26)(22,23)(24,29)(25,34)(27,30)(28,31)(32,33)(35,36)Polycyclic presentation:
2^9|a^2=fgh,b^2=fg,c^2=fg,d^2=hi,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, 1652988);