Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #1506685 |
| 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 | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2189 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)(7,11)(8,9)(10,12)(13,14,17,16)(15,19,20,18)(22,26)(24,31)(27,32)(30,35);(2,4)(6,9)(10,11)(22,26)(23,29)(28,33)(30,35);(1,2)(3,4)(7,12)(10,11)(21,22,25,26)(23,27,29,32)(24,28,31,33)(30,34,35,36);(5,7,8,12)(6,10,9,11)(13,15,17,20)(14,18,16,19)(21,23)(22,27)(24,30)(25,29)(26,32)(28,34)(31,35)(33,36);(5,6)(7,10)(8,9)(11,12)(13,16,17,14)(15,19,20,18)(21,24)(22,28)(23,30)(25,31)(26,33)(27,34)(29,35)(32,36)Polycyclic presentation:
2^9|a^2=fi,c^2=g,d^2=i,e^2=fi,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, 1743234);