Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4131229 |
| 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, prime radical, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2064 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 40 points:
(2,4)(9,10,14,12)(11,17,18,15)(13,20,21,16)(19,22,24,23)(25,26,30,34)(27,28,32,31)(29,37,40,33)(35,39,38,36);(6,8)(11,18)(13,21)(15,17)(16,20)(25,27,30,32)(26,31,34,28)(29,38,40,35)(33,36,37,39);(5,6)(7,8)(9,11,14,18)(10,15,12,17)(13,19,21,24)(16,22,20,23)(25,27)(26,31)(28,34)(29,38)(30,32)(33,36)(35,40)(37,39);(1,2)(3,4)(9,12,14,10)(11,15,18,17)(13,20,21,16)(19,22,24,23)(25,28,30,31)(26,32,34,27)(29,39,40,36)(33,35,37,38);(9,13)(10,16)(11,19)(12,20)(14,21)(15,22)(17,23)(18,24)(25,29,30,40)(26,33,34,37)(27,35,32,38)(28,36,31,39)Polycyclic presentation:
2^9|a^2=g,b^2=f,c^2=fg,d^2=g,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^b=eg,d^c=dg,e^c=ef,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4038667);