Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #6132643 |
| 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, cosocle | C24 | C25 |
| prime radical | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1449 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(2,5)(3,7)(11,15)(13,16)(17,18,22,19)(20,25,28,23)(21,24,29,26)(27,31,32,30)(33,34,38,41)(35,42,39,36)(37,40,45,47)(43,48,46,44);(2,5)(6,8)(9,10,12,14)(11,13,15,16)(18,19)(23,25)(24,26)(30,31)(35,39)(36,42)(37,45)(40,47);(9,10)(11,13)(12,14)(15,16)(17,19,22,18)(20,23,28,25)(21,24,29,26)(27,31,32,30)(33,35)(34,36)(37,43)(38,39)(40,44)(41,42)(45,46)(47,48);(1,2)(3,6)(4,5)(7,8)(17,20,22,28)(18,23,19,25)(21,27,29,32)(24,30,26,31)(33,36)(34,39)(35,41)(37,44)(38,42)(40,46)(43,47)(45,48);(1,3)(2,6)(4,7)(5,8)(9,11)(10,13)(12,15)(14,16)(17,21,22,29)(18,24,19,26)(20,27,28,32)(23,30,25,31)(33,37,38,45)(34,40,41,47)(35,43,39,46)(36,44,42,48)Polycyclic presentation:
2^9|a^2=fg,b^2=fgh,c^2=f,d^2=f,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=ehi,c^b=ch,d^b=dfi,e^b=eg,d^c=dfg,e^c=ef62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6042440);