Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2601293 |
| 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 | |
1908 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 36 points:
(5,6,8,9)(7,11,12,10)(13,14,17,15)(16,19,20,18)(21,22,26,31)(23,28,33,27)(24,29,34,36)(25,30,35,32);(2,4)(6,9)(10,11)(21,23,26,33)(22,27,31,28)(24,32,34,30)(25,29,35,36);(1,2)(3,4)(5,6)(7,10)(8,9)(11,12)(13,15,17,14)(16,19,20,18)(21,23)(22,28)(24,32)(25,29)(26,33)(27,31)(30,34)(35,36);(5,6)(7,11)(8,9)(10,12)(13,16)(14,18)(15,19)(17,20)(21,24)(22,29)(23,32)(25,28)(26,34)(27,35)(30,33)(31,36);(5,7,8,12)(6,10,9,11)(21,25)(22,30)(23,29)(24,27)(26,35)(28,34)(31,32)(33,36)Polycyclic presentation:
2^9|a^2=fgi,b^2=fg,c^2=gi,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dg,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2540668);