Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3069037 |
| 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, prime radical, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1542 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 40 points:
(2,4)(6,8)(9,10)(11,15)(12,14)(13,16)(17,18,23,21)(19,24,22,20)(25,26)(27,31)(28,36)(29,30)(32,38)(33,40)(34,35)(37,39);(3,7)(6,8)(9,11,14,13)(10,15,12,16)(17,19,23,22)(18,24,21,20)(26,29)(31,34)(32,36)(39,40);(1,2,5,4)(3,6,7,8)(9,11,14,13)(10,16,12,15)(17,20)(18,22)(19,21)(23,24)(25,27,30,35)(26,31,29,34)(28,37,38,33)(32,40,36,39);(1,3,5,7)(2,6,4,8)(9,12,14,10)(11,15,13,16)(17,21,23,18)(19,24,22,20)(25,28,30,38)(26,32,29,36)(27,33,35,37)(31,39,34,40);(1,4,5,2)(3,8,7,6)(9,13,14,11)(10,15,12,16)(17,22,23,19)(18,24,21,20)(25,29,30,26)(27,34,35,31)(28,32,38,36)(33,39,37,40)Polycyclic presentation:
2^9|a^2=h,b^2=fhi,c^2=fgh,d^2=fg,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dg,e^b=ef,d^c=df,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2997035);