Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4465744 |
| 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 | |
1944 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)(5,6,10,11)(7,12,9,8)(14,17)(15,19)(23,29)(24,32)(26,30)(27,34);(6,11)(7,9)(15,19)(18,20)(21,22,25,28)(23,30,29,26)(24,27,32,34)(31,36,35,33);(5,7,10,9)(6,12,11,8)(13,14)(15,18)(16,17)(19,20)(21,23,25,29)(22,26,28,30)(24,31,32,35)(27,33,34,36);(1,2)(3,4)(5,8,10,12)(6,7,11,9)(13,15)(14,18)(16,19)(17,20)(21,22)(23,26)(24,27)(25,28)(29,30)(31,33)(32,34)(35,36);(5,9,10,7)(6,12,11,8)(21,24)(22,27)(23,31)(25,32)(26,33)(28,34)(29,35)(30,36)Polycyclic presentation:
2^9|a^2=f,b^2=fh,c^2=h,d^2=f,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dg,e^b=ef,d^c=df68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4374615);