Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4131229 |
| 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 | |
1460 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(3,7)(6,8)(17,18)(19,22)(20,25)(21,26)(23,28)(24,29)(27,31)(30,32)(33,34)(35,38)(36,39)(37,40)(41,46)(42,47)(43,45)(44,48);(10,13)(14,16)(18,19)(21,29)(23,25)(27,30)(33,34,37,40)(35,38,42,47)(36,39,45,43)(41,46,44,48);(1,2,5,4)(3,6,7,8)(9,10)(11,14)(12,13)(15,16)(17,19,22,18)(20,23,28,25)(21,24,29,26)(27,32,30,31)(33,34)(35,38)(36,43)(37,40)(39,45)(41,48)(42,47)(44,46);(1,3)(2,6)(4,8)(5,7)(17,20,22,28)(18,23,19,25)(21,30,29,27)(24,32,26,31)(33,35,37,42)(34,38,40,47)(36,44,45,41)(39,48,43,46);(1,4,5,2)(3,8,7,6)(9,11,12,15)(10,14,13,16)(17,21)(18,24)(19,26)(20,27)(22,29)(23,31)(25,32)(28,30)(33,36)(34,39)(35,41)(37,45)(38,46)(40,43)(42,44)(47,48)Polycyclic presentation:
2^9|b^2=fg,c^2=i,d^2=f,e^2=fghi,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^b=eg,d^c=dg,e^c=ef,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4207947);