Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4033837 |
| 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 | C22×#121 | C22 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2320 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 36 points:
(6,7)(8,12)(13,14,18,16)(15,17,20,19)(21,22)(23,29)(24,27)(25,28)(26,31)(30,36)(32,34)(33,35);(2,4)(8,12)(10,11)(15,20)(17,19)(21,22,25,28)(23,29,31,26)(24,27,34,32)(30,36,33,35);(1,2,3,4)(5,6,9,7)(8,10,12,11)(13,15,18,20)(14,19,16,17)(21,22)(23,26)(24,32)(25,28)(27,34)(29,31)(30,36)(33,35);(5,7,9,6)(8,11,12,10)(13,16,18,14)(15,17,20,19)(21,23)(22,26)(24,33)(25,31)(27,36)(28,29)(30,34)(32,35);(5,8)(6,10)(7,11)(9,12)(13,17)(14,15)(16,20)(18,19)(21,24)(22,27)(23,30)(25,34)(26,35)(28,32)(29,36)(31,33)Polycyclic presentation:
2^9|a^2=fgi,b^2=f,c^2=hi,d^2=g,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=df,e^b=eg,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, 3992008);