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 | |
2454 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)(10,12)(14,17)(15,19)(21,22,26,27)(23,30,29,25)(24,28,33,35)(31,32,36,34);(5,6,9,8)(7,10,11,12)(14,17)(18,20)(22,27)(23,29)(28,35)(31,36);(1,2)(3,4)(13,14)(15,18)(16,17)(19,20)(21,23,26,29)(22,25,27,30)(24,31,33,36)(28,34,35,32);(5,7)(6,10)(8,12)(9,11)(21,24)(22,28)(23,31)(25,34)(26,33)(27,35)(29,36)(30,32);(1,2)(3,4)(5,8,9,6)(7,12,11,10)(13,15,16,19)(14,18,17,20)(21,25)(22,29)(23,27)(24,32)(26,30)(28,31)(33,34)(35,36)Polycyclic presentation:
2^9|a^2=fgh,b^2=i,c^2=f,e^2=fhi,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, 1268546);