Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #1258513 |
| 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 | C2×#392 | C22 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2250 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 36 points:
(2,3)(6,9)(10,12)(13,14,16,18)(15,19,20,17)(22,26)(23,29)(24,31)(35,36);(15,20)(17,19)(21,22,25,26)(23,27,29,33)(24,28,31,34)(30,35,32,36);(1,2,4,3)(13,15,16,20)(14,17,18,19)(21,22)(23,27)(24,28)(25,26)(29,33)(30,35)(31,34)(32,36);(5,6)(7,10)(8,9)(11,12)(21,23,25,29)(22,27,26,33)(24,32,31,30)(28,36,34,35);(1,3,4,2)(5,7,8,11)(6,10,9,12)(13,14)(15,17)(16,18)(19,20)(21,24,25,31)(22,28,26,34)(23,30,29,32)(27,35,33,36)Polycyclic presentation:
2^9|a^2=fh,b^2=f,c^2=fg,d^2=f,e^2=ghi,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^d=ef86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1244508);