Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #1393169 |
| 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 | |
1980 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)(6,7)(8,12)(21,22,26,24)(23,28,31,27)(25,29,33,32)(30,35,36,34);(14,17)(18,20)(21,23,26,31)(22,27,24,28)(25,30,33,36)(29,34,32,35);(5,6,9,7)(8,10,12,11)(13,14)(15,18)(16,17)(19,20)(21,23)(22,28)(24,27)(25,30)(26,31)(29,35)(32,34)(33,36);(1,2)(3,4)(5,7,9,6)(8,11,12,10)(21,24,26,22)(23,28,31,27)(25,29,33,32)(30,34,36,35);(5,8)(6,10)(7,11)(9,12)(13,15,16,19)(14,18,17,20)(21,25)(22,29)(23,30)(24,32)(26,33)(27,34)(28,35)(31,36)Polycyclic presentation:
2^9|a^2=f,b^2=f,c^2=g,d^2=fg,e^2=fh,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^c=df,e^d=ef86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1304016);