Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5956507 |
| 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 | |
1298 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)(3,7,8,5)(11,15)(14,16)(17,18,21,22)(19,23,25,29)(20,27,28,24)(26,32,31,30)(33,34,39,42)(35,40,36,37)(38,46,47,41)(43,45,44,48);(3,8)(5,7)(10,13)(14,16)(18,22)(19,25)(20,28)(30,32)(33,35,39,36)(34,40,42,37)(38,43,47,44)(41,48,46,45);(9,10)(11,14)(12,13)(15,16)(17,18)(19,23)(20,24)(21,22)(25,29)(26,30)(27,28)(31,32)(33,36,39,35)(34,40,42,37)(38,44,47,43)(41,48,46,45);(1,3)(2,5)(4,8)(6,7)(17,19,21,25)(18,23,22,29)(20,26,28,31)(24,30,27,32)(33,37,39,40)(34,36,42,35)(38,45,47,48)(41,44,46,43);(1,2)(3,5)(4,6)(7,8)(9,11,12,15)(10,14,13,16)(17,20,21,28)(18,24,22,27)(19,26,25,31)(23,30,29,32)(33,38,39,47)(34,41,42,46)(35,43,36,44)(37,45,40,48)Polycyclic presentation:
2^9|a^2=fi,b^2=fg,c^2=fg,d^2=g,e^2=fgh,b^a=bf,c^a=cg,d^a=di,e^a=ehi,c^b=ch,d^b=dgi,e^b=ef,d^c=dfg62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5910168);