Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3069037 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
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 | |
2346 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)(5,6,8,10)(7,11,12,9)(14,16)(15,19)(21,22)(23,29)(24,26)(25,28)(27,30)(31,35)(32,33)(34,36);(7,12)(9,11)(15,19)(18,20)(22,24)(23,30)(25,33)(34,35);(5,7)(6,9)(8,12)(10,11)(13,14,17,16)(15,18,19,20)(21,23)(22,27)(24,29)(25,31)(26,30)(28,34)(32,35)(33,36);(1,2)(3,4)(5,7)(6,9)(8,12)(10,11)(13,15)(14,18)(16,20)(17,19)(21,24,26,22)(23,27,30,29)(25,28,33,32)(31,35,36,34);(5,6)(7,9)(8,10)(11,12)(13,16,17,14)(15,20,19,18)(21,25,26,33)(22,28,24,32)(23,31,30,36)(27,34,29,35)Polycyclic presentation:
2^9|a^2=fh,c^2=gh,d^2=f,e^2=fgh,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dg,e^b=ef,d^c=df,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2988418);