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 | |
1308 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,5,4)(3,6,7,8)(9,10,13,11)(12,15,16,14)(18,22)(20,27)(23,28)(26,31)(33,34)(35,38)(36,39)(37,40)(41,43)(42,45)(44,47)(46,48);(3,7)(6,8)(9,11,13,10)(12,15,16,14)(19,25)(23,28)(26,31)(30,32)(34,35)(36,43)(37,45)(47,48);(1,3,5,7)(2,6,4,8)(17,18)(19,23)(20,24)(21,22)(25,28)(26,30)(27,29)(31,32)(33,35,38,34)(36,39,43,41)(37,42,45,40)(44,47,46,48);(1,2,5,4)(3,6,7,8)(9,12,13,16)(10,14,11,15)(17,19,21,25)(18,23,22,28)(20,26,27,31)(24,30,29,32)(33,36,38,43)(34,39,35,41)(37,46,45,44)(40,48,42,47);(1,4,5,2)(3,8,7,6)(17,20,21,27)(18,24,22,29)(19,26,25,31)(23,30,28,32)(33,37,38,45)(34,40,35,42)(36,44,43,46)(39,47,41,48)Polycyclic presentation:
2^9|a^2=fhi,b^2=i,c^2=h,d^2=fghi,e^2=fgh,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, 3075458);