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 | C2×#392 | C22 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2275 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)(15,19)(16,17)(18,20)(21,22)(23,29)(24,32)(25,26)(27,31)(28,33)(30,34)(35,36);(8,12)(10,11)(13,15,18,16)(14,19,20,17)(22,26)(24,33)(27,29)(30,36);(6,7)(10,11)(14,20)(15,16)(21,23,25,31)(22,27,26,29)(24,30,33,36)(28,34,32,35);(1,2,3,4)(5,7,9,6)(8,11,12,10)(13,16,18,15)(14,19,20,17)(21,24)(22,28)(23,30)(25,33)(26,32)(27,34)(29,35)(31,36);(5,8)(6,10)(7,11)(9,12)(13,17,18,19)(14,15,20,16)(21,22)(23,29)(24,28)(25,26)(27,31)(30,35)(32,33)(34,36)Polycyclic presentation:
2^9|b^2=h,c^2=f,d^2=fi,e^2=h,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, 4990271);