Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4465744 |
| 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)(10,14)(12,16)(17,18)(19,22)(20,25)(21,26)(23,27)(24,29)(28,30)(31,32)(33,34,39,37)(35,40,36,43)(38,41,46,45)(42,47,44,48);(10,14)(11,15)(18,19)(20,27)(21,29)(30,31)(33,35,39,36)(34,40,37,43)(38,42,46,44)(41,47,45,48);(1,2,5,4)(3,6,7,8)(9,10)(11,12)(13,14)(15,16)(17,19,22,18)(20,23,27,25)(21,26,29,24)(28,30,32,31)(33,36,39,35)(34,40,37,43)(38,44,46,42)(41,47,45,48);(1,3)(2,6)(4,8)(5,7)(9,11)(10,12)(13,15)(14,16)(17,20)(18,23)(19,25)(21,28)(22,27)(24,30)(26,31)(29,32)(33,37,39,34)(35,40,36,43)(38,45,46,41)(42,47,44,48);(1,4,5,2)(3,8,7,6)(9,12)(10,11)(13,16)(14,15)(17,21)(18,24)(19,26)(20,28)(22,29)(23,30)(25,31)(27,32)(33,38,39,46)(34,41,37,45)(35,42,36,44)(40,47,43,48)Polycyclic presentation:
2^9|a^2=gh,b^2=gh,c^2=ghi,d^2=gh,e^2=fghi,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dg,e^b=ef,d^c=df68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4505036);