Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5035983 |
| automorphism | Aut(G) order 1048576 |
| 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 | |
2042 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)(5,6)(7,9)(8,10)(11,12)(13,14,17,16)(15,19,20,18)(23,29)(25,33)(27,30)(28,32);(8,12)(10,11)(14,16)(15,20)(21,22,26,24)(23,30,29,27)(25,32,33,28)(31,34,35,36);(6,7)(10,11)(14,16)(18,19)(21,23,26,29)(22,27,24,30)(25,31,33,35)(28,34,32,36);(1,2)(3,4)(5,7,9,6)(8,11,12,10)(13,15,17,20)(14,18,16,19)(21,24,26,22)(23,27,29,30)(25,32,33,28)(31,34,35,36);(5,8)(6,10)(7,11)(9,12)(13,16,17,14)(15,19,20,18)(21,25)(22,28)(23,31)(24,32)(26,33)(27,34)(29,35)(30,36)Polycyclic presentation:
2^9|a^2=f,b^2=h,c^2=h,d^2=g,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,e^b=eg,d^c=dfg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4944352);