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)(10,13)(11,15)(17,18)(19,23)(20,27)(21,22)(24,29)(25,32)(26,30)(28,31)(33,34,38,36)(35,39,41,42)(37,44,45,40)(43,48,47,46);(1,2,5,4)(3,6,7,8)(11,15)(14,16)(18,22)(23,30)(24,27)(31,32)(35,41)(39,42)(43,47)(46,48);(1,2,5,4)(3,6,7,8)(9,10)(11,14)(12,13)(15,16)(17,18)(19,23)(20,24)(21,22)(25,31)(26,30)(27,29)(28,32)(33,35,38,41)(34,39,36,42)(37,43,45,47)(40,46,44,48);(1,3)(2,6)(4,8)(5,7)(9,11,12,15)(10,14,13,16)(17,19,21,26)(18,23,22,30)(20,28,29,25)(24,32,27,31)(33,36,38,34)(35,42,41,39)(37,44,45,40)(43,48,47,46);(1,4,5,2)(3,8,7,6)(17,20,21,29)(18,24,22,27)(19,25,26,28)(23,31,30,32)(33,37)(34,40)(35,43)(36,44)(38,45)(39,46)(41,47)(42,48)Polycyclic presentation:
2^9|a^2=fh,b^2=gi,c^2=fghi,d^2=gh,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, 4835401);