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 | |
1304 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)(5,8)(11,15)(13,16)(17,18,21,24)(19,22,25,28)(20,23,27,30)(26,29,31,32)(33,34,38,39)(35,36,43,40)(37,41,45,46)(42,44,48,47);(1,2,4,6)(3,5,7,8)(19,25)(22,28)(26,31)(29,32)(34,39)(36,40)(37,45)(42,48);(1,2)(3,5)(4,6)(7,8)(9,10,12,14)(11,13,15,16)(17,18)(19,22)(20,23)(21,24)(25,28)(26,29)(27,30)(31,32)(33,35,38,43)(34,40,39,36)(37,42,45,48)(41,47,46,44);(9,11,12,15)(10,13,14,16)(17,19)(18,22)(20,26)(21,25)(23,29)(24,28)(27,31)(30,32)(33,36,38,40)(34,35,39,43)(37,44,45,47)(41,42,46,48);(1,3)(2,5)(4,7)(6,8)(17,20,21,27)(18,23,24,30)(19,26,25,31)(22,29,28,32)(33,37)(34,41)(35,42)(36,44)(38,45)(39,46)(40,47)(43,48)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=i,d^2=i,e^2=fg,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, 4425356);