Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5211343 |
| 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 | |
1590 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 40 points:
(2,5)(6,8)(9,10)(11,12)(13,14)(15,16)(18,20)(22,23)(25,26,30,33)(27,28,31,36)(29,32,38,40)(34,35,37,39);(10,14)(11,16)(17,18)(19,23)(20,21)(22,24)(25,27,30,31)(26,28,33,36)(29,37,38,34)(32,39,40,35);(11,16)(12,15)(17,19,21,24)(18,22,20,23)(25,26)(27,28)(29,32)(30,33)(31,36)(34,35)(37,39)(38,40);(1,2)(3,6)(4,5)(7,8)(9,11,13,16)(10,15,14,12)(17,20,21,18)(19,22,24,23)(25,28)(26,31)(27,33)(29,35)(30,36)(32,37)(34,40)(38,39);(1,3,4,7)(2,6,5,8)(9,12,13,15)(10,16,14,11)(17,19)(18,23)(20,22)(21,24)(25,29)(26,32)(27,34)(28,35)(30,38)(31,37)(33,40)(36,39)Polycyclic presentation:
2^9|a^2=g,b^2=g,c^2=h,d^2=f,e^2=hi,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^b=df,e^b=eg,d^c=dfg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5270467);