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) = 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 | |
1418 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,5,8)(3,6,4,7)(9,10,13,15)(11,14,16,12)(17,18,21,20)(19,22,24,23)(25,26,30,32)(27,33,35,28)(29,37,38,31)(34,36,40,39);(1,3,5,4)(2,6,8,7)(9,10)(11,12)(13,15)(14,16)(17,19,21,24)(18,22,20,23)(29,38)(31,37)(34,40)(36,39);(9,11,13,16)(10,14,15,12)(17,19)(18,22)(20,23)(21,24)(25,27,30,35)(26,28,32,33)(29,34,38,40)(31,36,37,39);(1,2)(3,6)(4,7)(5,8)(9,10)(11,14)(12,16)(13,15)(17,18,21,20)(19,22,24,23)(25,28)(26,27)(29,36)(30,33)(31,34)(32,35)(37,40)(38,39);(1,4,5,3)(2,7,8,6)(9,12,13,14)(10,11,15,16)(17,20,21,18)(19,23,24,22)(25,29,30,38)(26,31,32,37)(27,34,35,40)(28,36,33,39)Polycyclic presentation:
2^9|a^2=fghi,b^2=hi,c^2=fg,d^2=fh,e^2=fghi,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, 4206478);