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:
(1,2,4,6)(3,5,7,8)(10,13)(14,16)(20,27)(24,29)(25,31)(30,32)(33,34,39,36)(35,41,37,40)(38,44,46,42)(43,47,45,48);(11,15)(14,16)(18,22)(23,28)(24,29)(30,32)(33,35,39,37)(34,40,36,41)(38,45,46,43)(42,48,44,47);(9,10)(11,14)(12,13)(15,16)(17,18,21,22)(19,23,26,28)(20,24,27,29)(25,30,31,32)(33,36,39,34)(35,40,37,41)(38,44,46,42)(43,47,45,48);(1,2)(3,5)(4,6)(7,8)(9,11,12,15)(10,14,13,16)(17,19,21,26)(18,23,22,28)(20,25,27,31)(24,30,29,32)(33,37,39,35)(34,41,36,40)(38,45,46,43)(42,48,44,47);(1,3,4,7)(2,5,6,8)(17,20)(18,24)(19,25)(21,27)(22,29)(23,30)(26,31)(28,32)(33,38)(34,42)(35,43)(36,44)(37,45)(39,46)(40,47)(41,48)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=h,d^2=gh,e^2=fi,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, 4462078);