Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #1506685 |
| automorphism | ? |
| 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 | |
2370 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)(9,10,14,15)(11,12,19,16)(13,21,22,17)(18,23,24,20)(25,26,29,33)(27,31,35,30)(28,32,37,40)(34,39,36,38);(6,8)(10,15)(12,16)(13,22)(18,24)(25,27,29,35)(26,30,33,31)(28,36,37,34)(32,39,40,38);(5,6)(7,8)(9,11,14,19)(10,16,15,12)(13,18,22,24)(17,23,21,20)(25,27)(26,31)(28,34)(29,35)(30,33)(32,39)(36,37)(38,40);(1,2,3,4)(9,12,14,16)(10,11,15,19)(13,20,22,23)(17,18,21,24)(25,26)(27,30)(28,32)(29,33)(31,35)(34,38)(36,39)(37,40);(9,13,14,22)(10,17,15,21)(11,18,19,24)(12,20,16,23)(25,28)(26,32)(27,34)(29,37)(30,38)(31,39)(33,40)(35,36)Polycyclic presentation:
2^9|a^2=f,b^2=fg,c^2=g,d^2=fgi,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dg,e^b=ef,d^c=df74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1455278);