Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5956507 |
| automorphism | Aut(G) order 2097152 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C2×#392 | C22 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1848 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(2,3)(4,8)(9,10,14,12)(11,16,13,15)(18,20)(19,23)(25,26,29,32)(27,33,35,30)(28,36,37,31)(34,38,40,39);(1,2,5,3)(4,7,8,6)(10,12)(11,13)(17,18,22,20)(19,24,23,21);(1,2,5,3)(4,6,8,7)(9,11,14,13)(10,15,12,16)(17,19)(18,21)(20,24)(22,23)(25,26)(27,33)(28,31)(29,32)(30,35)(34,39)(36,37)(38,40);(1,3,5,2)(4,7,8,6)(9,12,14,10)(11,16,13,15)(17,20,22,18)(19,24,23,21)(25,27,29,35)(26,30,32,33)(28,34,37,40)(31,38,36,39);(1,4,5,8)(2,6,3,7)(9,13,14,11)(10,16,12,15)(17,21,22,24)(18,23,20,19)(25,28,29,37)(26,31,32,36)(27,34,35,40)(30,38,33,39)Polycyclic presentation:
2^9|a^2=fi,b^2=fgi,c^2=fghi,d^2=g,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=ehi,c^b=ch,d^b=dgi,e^b=ef,d^c=dfg62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5959020);