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 | |
1832 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 36 points:
(5,6,8,10)(7,11,12,9)(13,14,17,15)(16,19,20,18)(22,26)(23,29)(24,32)(35,36);(2,4)(7,12)(9,11)(21,22,25,26)(23,30,29,27)(24,28,32,34)(31,36,33,35);(1,2)(3,4)(5,6)(7,9)(8,10)(11,12)(13,15,17,14)(16,19,20,18)(21,22)(23,27)(24,28)(25,26)(29,30)(31,35)(32,34)(33,36);(5,7,8,12)(6,9,10,11)(13,16,17,20)(14,18,15,19)(21,23,25,29)(22,27,26,30)(24,33,32,31)(28,36,34,35);(5,6)(7,9)(8,10)(11,12)(21,24)(22,28)(23,31)(25,32)(26,34)(27,35)(29,33)(30,36)Polycyclic presentation:
2^9|a^2=fi,b^2=f,c^2=gi,d^2=i,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, 2569348);