Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2760365 |
| 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 | |
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)(11,15)(12,20)(13,14)(16,21)(17,18)(19,24)(22,23)(26,27)(28,35)(32,34)(36,37);(6,8)(10,14)(11,17)(16,20)(19,23)(25,26,30,27)(28,31,35,33)(29,32,38,34)(36,39,37,40);(5,6)(7,8)(9,11,13,17)(10,15,14,18)(12,19,21,23)(16,22,20,24)(25,27,30,26)(28,31,35,33)(29,34,38,32)(36,39,37,40);(1,2)(3,4)(9,10)(11,18)(12,20)(13,14)(15,17)(16,21)(19,22)(23,24)(25,28,30,35)(26,31,27,33)(29,37,38,36)(32,40,34,39);(9,12,13,21)(10,16,14,20)(11,19,17,23)(15,22,18,24)(25,29,30,38)(26,32,27,34)(28,36,35,37)(31,39,33,40)Polycyclic presentation:
2^9|b^2=g,c^2=f,d^2=g,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dfg,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, 2667693);