Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5605735 |
| 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 | |
1386 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)(9,10,12,14)(11,13,15,16)(17,18)(19,22)(20,25)(21,24)(23,28)(26,29)(27,32)(30,31)(33,34,39,41)(35,37,36,43)(38,40,46,48)(42,45,44,47);(2,5)(6,8)(11,15)(13,16)(18,19)(21,29)(23,25)(27,30)(33,35,39,36)(34,37,41,43)(38,44,46,42)(40,47,48,45);(1,2)(3,6)(4,5)(7,8)(9,11,12,15)(10,13,14,16)(17,19,22,18)(20,25,28,23)(21,26,29,24)(27,32,30,31)(33,36,39,35)(34,37,41,43)(38,44,46,42)(40,45,48,47);(9,10)(11,13)(12,14)(15,16)(17,20,22,28)(18,23,19,25)(21,30,29,27)(24,32,26,31)(33,37)(34,36)(35,41)(38,45)(39,43)(40,44)(42,48)(46,47);(1,3)(2,6)(4,7)(5,8)(9,10)(11,13)(12,14)(15,16)(17,21)(18,24)(19,26)(20,27)(22,29)(23,31)(25,32)(28,30)(33,38,39,46)(34,40,41,48)(35,42,36,44)(37,45,43,47)Polycyclic presentation:
2^9|a^2=gi,b^2=fg,c^2=fgi,d^2=f,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=eg,d^c=dfg,e^d=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5622945);