Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5024635 |
| automorphism | Aut(G) order 2097152 |
| 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 | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2634 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)(5,6,9,10)(7,8,11,12)(15,19)(18,20)(21,22)(23,26)(24,29)(25,30)(27,32)(28,34)(31,35)(33,36);(5,7)(6,8)(9,11)(10,12)(14,17)(18,20)(22,23)(27,29)(28,30)(35,36);(6,10)(7,11)(13,14)(15,18)(16,17)(19,20)(21,23,26,22)(24,27,32,29)(25,28,34,30)(31,36,33,35);(1,2)(3,4)(5,7)(6,8)(9,11)(10,12)(21,24,26,32)(22,27,23,29)(25,33,34,31)(28,36,30,35);(5,8)(6,7)(9,12)(10,11)(13,15)(14,18)(16,19)(17,20)(21,25,26,34)(22,28,23,30)(24,31,32,33)(27,35,29,36)Polycyclic presentation:
2^9|a^2=gi,c^2=f,d^2=f,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^c=dg,e^c=ef,e^d=ef80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4933000);