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 | |
1548 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,4,6)(3,5,7,8)(11,15)(12,16)(19,23)(21,24)(25,26)(27,28)(29,33)(30,31)(32,35)(34,36)(37,39)(38,40);(3,7)(5,8)(9,10,13,14)(11,16,15,12)(26,31)(28,32)(29,37)(34,40);(1,2)(3,5)(4,6)(7,8)(9,11)(10,12)(13,15)(14,16)(17,18,20,22)(19,21,23,24)(25,27,30,35)(26,32,31,28)(29,34,37,40)(33,38,39,36);(1,3)(2,5)(4,7)(6,8)(9,10)(11,12)(13,14)(15,16)(17,19,20,23)(18,21,22,24)(25,28,30,32)(26,27,31,35)(29,38,37,36)(33,40,39,34);(1,2)(3,5)(4,6)(7,8)(9,12)(10,11)(13,16)(14,15)(25,29,30,37)(26,33,31,39)(27,34,35,40)(28,36,32,38)Polycyclic presentation:
2^9|a^2=fgh,b^2=h,c^2=fi,d^2=fi,e^2=f,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, 3077182);