Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4597856 |
| 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 | |
2291 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)(7,11)(9,12)(13,14,17,19)(15,18,16,20)(21,22)(23,26)(24,27)(25,28)(29,31)(30,33)(32,34)(35,36);(5,6,8,10)(7,9,11,12)(13,15,17,16)(14,18,19,20)(22,23)(24,31)(28,30)(32,36);(5,6)(7,9)(8,10)(11,12)(13,16,17,15)(14,18,19,20)(21,23,26,22)(24,29,31,27)(25,28,33,30)(32,34,36,35);(1,2)(3,4)(13,14)(15,20)(16,18)(17,19)(21,24,26,31)(22,27,23,29)(25,32,33,36)(28,34,30,35);(5,7)(6,9)(8,11)(10,12)(21,25)(22,28)(23,30)(24,32)(26,33)(27,34)(29,35)(31,36)Polycyclic presentation:
2^9|a^2=i,b^2=gh,c^2=g,d^2=f,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^b=dg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4514106);