Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #1258513 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, prime radical, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1908 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 36 points:
(7,8)(9,11)(14,18)(16,20)(21,22,26,27)(23,28,30,24)(25,29,34,35)(31,33,36,32);(2,4)(5,6,10,12)(7,11,8,9)(22,27)(23,30)(29,35)(31,36);(1,2)(3,4)(5,7,10,8)(6,11,12,9)(13,14,17,18)(15,16,19,20)(21,23)(22,28)(24,27)(25,31)(26,30)(29,33)(32,35)(34,36);(5,8,10,7)(6,9,12,11)(13,15,17,19)(14,16,18,20)(21,24,26,28)(22,23,27,30)(25,33,34,32)(29,36,35,31);(5,9)(6,7)(8,12)(10,11)(13,16)(14,19)(15,18)(17,20)(21,25,26,34)(22,29,27,35)(23,31,30,36)(24,32,28,33)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=g,d^2=fg,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^d=ef86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1176070);