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, cosocle | C24 | C25 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1817 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,5,6)(3,7,8,4)(9,10,14,16)(11,12,13,15)(18,22)(20,24)(25,26,29,32)(27,33,35,30)(28,36,37,31)(34,38,40,39);(2,6)(4,7)(9,11)(10,15)(12,16)(13,14)(17,18,21,22)(19,20,23,24)(28,37)(31,36)(34,40)(38,39);(1,3,5,8)(2,7,6,4)(9,12,14,15)(10,13,16,11)(17,19,21,23)(18,20,22,24)(25,27,29,35)(26,30,32,33)(28,34,37,40)(31,38,36,39);(1,4)(2,3)(5,7)(6,8)(9,13,14,11)(10,12,16,15)(17,20,21,24)(18,19,22,23)(25,28,29,37)(26,31,32,36)(27,34,35,40)(30,38,33,39);(1,3)(2,7)(4,6)(5,8)(9,12)(10,13)(11,16)(14,15)(17,19)(18,20)(21,23)(22,24)(25,26)(27,30)(28,31)(29,32)(33,35)(34,38)(36,37)(39,40)Polycyclic presentation:
2^9|a^2=fhi,b^2=ghi,c^2=fg,d^2=i,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, 3422206);