Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2601293 |
| 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 | |
1464 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)(17,18,21,22)(19,23,25,30)(20,27,29,24)(26,32,28,31)(33,34,37,40)(35,41,42,38)(36,39,44,46)(43,47,48,45);(1,2,5,4)(3,6,7,8)(10,13)(14,16)(18,22)(19,25)(24,27)(26,28)(35,42)(36,44)(38,41)(39,46);(9,10)(11,14)(12,13)(15,16)(17,18)(19,23)(20,24)(21,22)(25,30)(26,31)(27,29)(28,32)(33,35,37,42)(34,38,40,41)(36,43,44,48)(39,45,46,47);(1,3)(2,6)(4,8)(5,7)(17,19)(18,23)(20,28)(21,25)(22,30)(24,32)(26,29)(27,31)(33,36)(34,39)(35,43)(37,44)(38,45)(40,46)(41,47)(42,48);(1,4,5,2)(3,8,7,6)(9,11,12,15)(10,14,13,16)(17,20)(18,24)(19,26)(21,29)(22,27)(23,31)(25,28)(30,32)(33,34)(35,38)(36,39)(37,40)(41,42)(43,45)(44,46)(47,48)Polycyclic presentation:
2^9|a^2=g,b^2=i,c^2=fg,e^2=ghi,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dg,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2627437);