Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4861280 |
| 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 | |
1300 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 48 points:
(3,7)(6,8)(9,10)(11,14)(12,16)(13,15)(17,18,21,22)(19,25,26,23)(20,28,30,24)(27,31,29,32)(33,34,37,40)(35,38,41,45)(36,43,44,39)(42,48,47,46);(1,2,5,4)(3,6,7,8)(9,11,13,16)(10,14,15,12)(18,22)(19,26)(24,28)(27,29)(35,41)(38,45)(42,47)(46,48);(9,12)(10,11)(13,14)(15,16)(17,18)(19,23)(20,24)(21,22)(25,26)(27,31)(28,30)(29,32)(33,35,37,41)(34,38,40,45)(36,42,44,47)(39,46,43,48);(1,3)(2,6)(4,8)(5,7)(9,11)(10,12)(13,16)(14,15)(17,19,21,26)(18,23,22,25)(20,29,30,27)(24,32,28,31)(33,34)(35,38)(36,39)(37,40)(41,45)(42,46)(43,44)(47,48);(1,4,5,2)(3,8,7,6)(17,20,21,30)(18,24,22,28)(19,27,26,29)(23,31,25,32)(33,36,37,44)(34,39,40,43)(35,42,41,47)(38,46,45,48)Polycyclic presentation:
2^9|a^2=h,b^2=hi,c^2=fh,d^2=f,e^2=fgi,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dg,e^d=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4825074);