Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #1506685 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C25×C4 |
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 | |
1442 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)(3,7)(4,5)(6,8)(17,18,22,20)(19,23,21,24)(25,26,30,29)(27,31,35,34)(28,36,38,32)(33,40,37,39);(2,5)(3,8)(10,13)(14,16)(17,19,22,21)(18,23,20,24)(26,29)(31,34)(32,36)(39,40);(2,5)(6,7)(9,10,12,13)(11,14,15,16)(17,20,22,18)(19,23,21,24)(25,27,30,35)(26,31,29,34)(28,37,38,33)(32,40,36,39);(1,2)(3,6)(4,5)(7,8)(9,11,12,15)(10,14,13,16)(17,21,22,19)(18,23,20,24)(25,28,30,38)(26,32,29,36)(27,33,35,37)(31,39,34,40);(1,3)(2,6)(4,8)(5,7)(25,29,30,26)(27,34,35,31)(28,36,38,32)(33,40,37,39)Polycyclic presentation:
2^9|a^2=i,b^2=fgi,c^2=fgh,d^2=fgh,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dg,e^b=ef,d^c=df74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1724012);