Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3069037 |
| automorphism | Aut(G) order 2097152 |
| 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 | |
2298 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)(9,10,14,11)(12,17,20,15)(13,18,21,16)(19,23,22,24)(25,26)(27,33)(28,32)(29,30)(31,34)(35,39)(36,37)(38,40);(6,8)(10,11)(13,21)(15,17)(19,22)(26,30)(27,34)(28,36)(38,39);(5,6,7,8)(9,11,14,10)(12,15,20,17)(13,18,21,16)(19,23,22,24)(25,27)(26,31)(28,35)(29,34)(30,33)(32,38)(36,40)(37,39);(1,2)(3,4)(9,12,14,20)(10,15,11,17)(13,22,21,19)(16,24,18,23)(25,26)(27,33)(28,37)(29,30)(31,34)(32,36)(35,38)(39,40);(9,13,14,21)(10,16,11,18)(12,19,20,22)(15,23,17,24)(25,28)(26,32)(27,35)(29,36)(30,37)(31,38)(33,39)(34,40)Polycyclic presentation:
2^9|a^2=fg,c^2=gh,d^2=fg,e^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, 2978780);