Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3332717 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26 |
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 | |
1512 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 40 points:
(3,7)(6,8)(9,10,13,14)(11,15,16,12)(17,18,22,24)(19,20,21,23)(25,26)(27,30)(28,36)(29,33)(31,35)(32,37)(34,39)(38,40);(10,14)(12,15)(17,19)(18,23)(20,24)(21,22)(25,27,29,35)(26,30,33,31)(28,34,37,40)(32,38,36,39);(1,2,5,4)(3,6,7,8)(9,11,13,16)(10,15,14,12)(17,20,22,23)(18,21,24,19)(25,27)(26,31)(28,34)(29,35)(30,33)(32,39)(36,38)(37,40);(1,3,5,7)(2,6,4,8)(9,12)(10,11)(13,15)(14,16)(17,21,22,19)(18,20,24,23)(25,26)(27,31)(28,32)(29,33)(30,35)(34,39)(36,37)(38,40);(1,4,5,2)(3,8,7,6)(9,11)(10,15)(12,14)(13,16)(17,20)(18,21)(19,24)(22,23)(25,28,29,37)(26,32,33,36)(27,34,35,40)(30,38,31,39)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=fghi,d^2=i,e^2=hi,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,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, 3720366);