Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2601293 |
| 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 | |
1644 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2)(3,6)(4,7)(5,8)(11,15)(14,16)(17,18)(19,24)(20,22)(21,23)(25,26,30,31)(27,34,33,29)(28,36,37,32)(35,39,40,38);(1,3)(2,4)(5,7)(6,8)(9,10,13,12)(11,14,15,16)(17,19,21,20)(18,22,23,24)(26,31)(27,33)(28,37)(38,39);(17,18)(19,24)(20,22)(21,23)(25,27,30,33)(26,29,31,34)(28,35,37,40)(32,39,36,38);(1,3)(2,6)(4,8)(5,7)(9,11)(10,14)(12,16)(13,15)(17,20,21,19)(18,22,23,24)(25,28,30,37)(26,32,31,36)(27,35,33,40)(29,39,34,38);(1,4,5,6)(2,7,8,3)(9,12,13,10)(11,16,15,14)(17,18)(19,22)(20,24)(21,23)(25,29)(26,33)(27,31)(28,38)(30,34)(32,35)(36,40)(37,39)Polycyclic presentation:
2^9|a^2=g,b^2=hi,c^2=g,d^2=h,e^2=fghi,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dg,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2637689);