Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3069037 |
| 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, prime radical, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1504 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,5,6)(3,7,8,4)(9,10,12,14)(11,15,16,13)(25,26,31,34)(27,29,28,32)(30,33,39,40)(35,37,36,38);(1,3)(2,4)(5,8)(6,7)(9,10)(11,15)(12,14)(13,16)(18,22)(23,24)(25,27,31,28)(26,32,34,29)(30,36,39,35)(33,37,40,38);(3,8)(4,7)(9,10,12,14)(11,13,16,15)(17,18)(19,23)(20,24)(21,22)(25,28,31,27)(26,29,34,32)(30,36,39,35)(33,37,40,38);(1,4)(2,3)(5,7)(6,8)(9,10)(11,15)(12,14)(13,16)(17,19,21,20)(18,23,22,24)(25,29)(26,28)(27,34)(30,38)(31,32)(33,35)(36,40)(37,39);(1,2)(3,7)(4,8)(5,6)(9,11,12,16)(10,13,14,15)(17,20,21,19)(18,24,22,23)(25,30)(26,33)(27,35)(28,36)(29,37)(31,39)(32,38)(34,40)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=gi,d^2=fgh,e^2=fghi,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, 3167170);