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 | |
2108 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 36 points:
(6,8)(7,11)(14,17)(15,19)(21,22,26,29)(23,28,32,27)(24,33,35,25)(30,31,34,36);(2,4)(14,17)(18,20)(21,23,26,32)(22,27,29,28)(24,30,35,34)(25,31,33,36);(1,2,3,4)(5,6,9,8)(7,10,11,12)(13,14,16,17)(15,18,19,20)(21,23)(22,28)(24,34)(25,31)(26,32)(27,29)(30,35)(33,36);(5,7)(6,10)(8,12)(9,11)(21,24,26,35)(22,25,29,33)(23,30,32,34)(27,31,28,36);(5,8,9,6)(7,12,11,10)(13,15,16,19)(14,18,17,20)(21,25)(22,24)(23,31)(26,33)(27,30)(28,34)(29,35)(32,36)Polycyclic presentation:
2^9|a^2=f,b^2=f,c^2=hi,d^2=f,e^2=g,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, 1310860);