Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2735949 |
| automorphism | Aut(G) order 1048576 |
| 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 | |
1944 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)(13,14,17,15)(16,19,20,18)(21,22,26,23)(24,29,31,27)(25,28,34,30)(32,36,33,35);(2,4)(5,6,9,8)(7,11,12,10)(22,23)(24,31)(28,30)(32,33);(1,2)(3,4)(5,7,9,12)(6,10,8,11)(13,15,17,14)(16,19,20,18)(21,23,26,22)(24,27,31,29)(25,30,34,28)(32,35,33,36);(5,7)(6,11)(8,10)(9,12)(13,16,17,20)(14,18,15,19)(21,24,26,31)(22,27,23,29)(25,33,34,32)(28,36,30,35);(5,8,9,6)(7,11,12,10)(21,25,26,34)(22,28,23,30)(24,32,31,33)(27,35,29,36)Polycyclic presentation:
2^9|a^2=i,b^2=g,c^2=gi,d^2=i,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,d^c=df,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2654405);