Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #45438 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, prime radical, cosocle | C24 | C25 |
| commutator | C23 | C24×C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1958 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 36 points:
(1,2,3,4)(5,6,8,9)(7,10,11,12)(13,14)(15,19)(16,18)(17,20)(23,29)(24,32)(26,30)(27,34);(6,9)(7,11)(14,16)(15,17)(21,22,25,28)(23,30,29,26)(24,27,32,34)(31,36,35,33);(13,15,18,17)(14,19,16,20)(21,23,25,29)(22,26,28,30)(24,31,32,35)(27,33,34,36);(5,6)(7,10)(8,9)(11,12)(13,16,18,14)(15,20,17,19)(21,22)(23,26)(24,27)(25,28)(29,30)(31,33)(32,34)(35,36);(5,7,8,11)(6,10,9,12)(13,17,18,15)(14,20,16,19)(21,24)(22,27)(23,31)(25,32)(26,33)(28,34)(29,35)(30,36)Polycyclic presentation:
2^9|a^2=i,b^2=g,c^2=h,d^2=gh,e^2=f,b^a=bf,c^a=cg,d^a=df,e^a=eg,c^b=ch,d^b=dfg,e^b=ef104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 464475);