Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4414082 |
| 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 | |
2138 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 36 points:
(2,4)(7,11)(10,12)(15,18)(16,20)(21,22,26,27)(23,28,30,24)(25,29,34,35)(31,36,33,32);(6,9)(10,12)(13,14,17,19)(15,16,18,20)(22,27)(23,30)(29,35)(31,33);(5,6)(7,10)(8,9)(11,12)(13,15,17,18)(14,16,19,20)(21,23)(22,28)(24,27)(25,33)(26,30)(29,32)(31,34)(35,36);(1,2)(3,4)(13,16)(14,18)(15,19)(17,20)(21,24,26,28)(22,23,27,30)(25,32,34,36)(29,31,35,33);(5,7)(6,10)(8,11)(9,12)(21,25)(22,29)(23,31)(24,32)(26,34)(27,35)(28,36)(30,33)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=g,d^2=f,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^c=ef80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4321341);