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, cosocle | C24 | C25 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1586 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2)(3,6)(4,5)(7,8)(9,10)(11,16)(12,15)(13,14)(18,21)(19,20)(25,26)(27,33)(28,31)(29,32)(30,35)(34,39)(36,37)(38,40);(2,4)(3,7)(10,14)(11,12)(17,18)(19,23)(20,24)(21,22)(25,26,29,32)(27,33,35,30)(28,31,37,36)(34,39,40,38);(2,4)(6,8)(9,11,13,12)(10,15,14,16)(17,19,22,20)(18,23,21,24)(25,26)(27,33)(28,36)(29,32)(30,35)(31,37)(34,38)(39,40);(1,3)(2,6)(4,8)(5,7)(9,12,13,11)(10,15,14,16)(17,20,22,19)(18,23,21,24)(25,27,29,35)(26,30,32,33)(28,34,37,40)(31,38,36,39);(1,4,5,2)(3,8,7,6)(9,10)(11,16)(12,15)(13,14)(17,21,22,18)(19,23,20,24)(25,28,29,37)(26,31,32,36)(27,34,35,40)(30,38,33,39)Polycyclic presentation:
2^9|b^2=fg,c^2=hi,d^2=fghi,e^2=gh,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, 5246629);