Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3069037 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
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 | |
2200 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(2,4)(9,10,14,16)(11,17,18,12)(13,21,22,15)(19,20,24,23)(25,26)(27,33)(28,36)(29,32)(30,35)(31,38)(34,39)(37,40);(6,8)(11,18)(12,17)(19,24)(20,23)(25,26,29,32)(27,30,35,33)(28,36,38,31)(34,40,37,39);(5,6,7,8)(9,11)(10,12)(13,19)(14,18)(15,20)(16,17)(21,23)(22,24)(25,26)(27,33)(28,31)(29,32)(30,35)(34,40)(36,38)(37,39);(1,2)(3,4)(9,12)(10,11)(13,20)(14,17)(15,19)(16,18)(21,24)(22,23)(25,27)(26,30)(28,37)(29,35)(31,40)(32,33)(34,38)(36,39);(9,13,14,22)(10,15,16,21)(11,19,18,24)(12,20,17,23)(25,28)(26,31)(27,34)(29,38)(30,39)(32,36)(33,40)(35,37)Polycyclic presentation:
2^9|a^2=g,b^2=f,c^2=fgh,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dg,e^b=ef,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, 2978624);