Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #1506685 |
| automorphism | ? |
| 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 | |
1950 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 36 points:
(5,6,8,10)(7,9,11,12)(14,16)(18,20)(21,22,26,25)(23,27,31,30)(24,32,34,28)(29,36,33,35);(1,2,3,4)(15,19)(18,20)(22,25)(27,30)(28,32)(35,36);(1,2)(3,4)(13,14,17,16)(15,18,19,20)(21,23,26,31)(22,27,25,30)(24,33,34,29)(28,36,32,35);(5,6)(7,9)(8,10)(11,12)(13,15)(14,18)(16,20)(17,19)(21,24,26,34)(22,28,25,32)(23,29,31,33)(27,35,30,36);(5,7,8,11)(6,9,10,12)(13,16,17,14)(15,20,19,18)(21,25,26,22)(23,30,31,27)(24,32,34,28)(29,36,33,35)Polycyclic presentation:
2^9|a^2=i,b^2=h,c^2=fg,d^2=f,e^2=gi,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dg,e^b=ef,d^c=df74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1694227);