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, cosocle | C24 | C25 |
| prime radical | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1396 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)(9,10,13,16)(11,14,15,12)(17,18)(19,22)(20,25)(21,24)(23,28)(26,29)(27,31)(30,32)(33,34,38,36)(35,41,43,39)(37,40,46,44)(42,48,45,47);(2,5)(6,8)(11,15)(12,14)(18,19)(21,29)(23,25)(27,32)(33,35,38,43)(34,39,36,41)(37,45,46,42)(40,48,44,47);(9,11,13,15)(10,14,16,12)(17,19,22,18)(20,25,28,23)(21,26,29,24)(27,31,32,30)(33,36,38,34)(35,39,43,41)(37,44,46,40)(42,47,45,48);(1,2)(3,6)(4,5)(7,8)(9,12,13,14)(10,15,16,11)(17,20)(18,23)(19,25)(21,27)(22,28)(24,30)(26,31)(29,32)(33,35)(34,39)(36,41)(37,42)(38,43)(40,47)(44,48)(45,46);(1,3)(2,6)(4,7)(5,8)(17,21)(18,24)(19,26)(20,27)(22,29)(23,30)(25,31)(28,32)(33,37,38,46)(34,40,36,44)(35,42,43,45)(39,47,41,48)Polycyclic presentation:
2^9|a^2=gh,b^2=fg,c^2=h,d^2=fh,e^2=fg,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, 5877025);