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 | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2075 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)(5,6,10,12)(7,8,11,9)(13,14)(15,19)(16,18)(17,20)(21,22)(23,31)(24,29)(25,33)(26,27)(28,32)(30,36)(34,35);(5,7,10,11)(6,8,12,9)(13,15,18,17)(14,19,16,20)(22,27)(23,32)(29,34)(30,33);(1,2,4,3)(5,8,10,9)(6,7,12,11)(13,16,18,14)(15,19,17,20)(21,23)(22,28)(24,33)(25,34)(26,32)(27,31)(29,36)(30,35);(5,6)(7,8)(9,11)(10,12)(13,15,18,17)(14,20,16,19)(21,24)(22,29)(23,33)(25,31)(26,35)(27,34)(28,36)(30,32);(1,3,4,2)(5,9,10,8)(6,11,12,7)(13,17,18,15)(14,19,16,20)(21,25,26,36)(22,30,27,33)(23,34,32,29)(24,28,35,31)Polycyclic presentation:
2^9|a^2=fhi,b^2=i,c^2=fg,d^2=fh,e^2=g,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, 1186644);