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 | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2352 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)(9,10,13,14)(11,17,18,15)(12,16,20,22)(19,24,21,23)(25,26)(27,32)(28,34)(29,37)(30,31)(33,38)(35,36)(39,40);(6,8)(10,14)(11,18)(12,20)(23,24)(26,31)(27,34)(29,38)(36,39);(5,6,7,8)(9,10)(11,17)(12,16)(13,14)(15,18)(19,24)(20,22)(21,23)(25,27,30,34)(26,32,31,28)(29,35,38,40)(33,39,37,36);(1,2)(3,4)(9,11)(10,15)(12,21)(13,18)(14,17)(16,24)(19,20)(22,23)(25,28,30,32)(26,27,31,34)(29,39,38,36)(33,40,37,35);(9,12)(10,16)(11,19)(13,20)(14,22)(15,23)(17,24)(18,21)(25,29)(26,33)(27,35)(28,36)(30,38)(31,37)(32,39)(34,40)Polycyclic presentation:
2^9|a^2=g,c^2=gh,d^2=fg,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, 2978513);