Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #1258513 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
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 | |
2232 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)(6,9)(7,11)(21,22,26,30)(23,29,25,27)(24,32,33,28)(31,35,34,36);(14,17)(18,20)(21,23,26,25)(22,27,30,29)(24,31,33,34)(28,35,32,36);(5,6)(7,10)(8,9)(11,12)(13,14,16,17)(15,18,19,20)(21,22)(23,27)(24,28)(25,29)(26,30)(31,35)(32,33)(34,36);(1,2)(3,4)(5,6)(7,10)(8,9)(11,12)(21,24)(22,28)(23,31)(25,34)(26,33)(27,35)(29,36)(30,32);(5,7,8,11)(6,10,9,12)(13,15,16,19)(14,18,17,20)(21,25,26,23)(22,29,30,27)(24,31,33,34)(28,35,32,36)Polycyclic presentation:
2^9|a^2=f,b^2=f,c^2=h,e^2=gh,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, 1174416);