Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4465744 |
| 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 | |
1459 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,4,6)(3,5,7,8)(9,10,12,14)(11,13,15,16)(17,18)(19,25)(20,28)(21,22)(23,27)(24,29)(26,30)(31,32)(34,38)(35,41)(36,44)(46,48);(3,7)(5,8)(11,15)(13,16)(18,22)(20,29)(23,25)(26,32)(33,34,37,38)(35,42,41,39)(36,45,44,40)(43,46,47,48);(1,3)(2,5)(4,7)(6,8)(9,11,12,15)(10,13,14,16)(17,18)(19,25)(20,24)(21,22)(23,27)(26,31)(28,29)(30,32)(33,34)(35,42)(36,40)(37,38)(39,41)(43,48)(44,45)(46,47);(9,10)(11,13)(12,14)(15,16)(17,19,21,27)(18,23,22,25)(20,26,29,32)(24,30,28,31)(33,35,37,41)(34,39,38,42)(36,43,44,47)(40,46,45,48);(1,2)(3,5)(4,6)(7,8)(9,10)(11,13)(12,14)(15,16)(17,20)(18,24)(19,26)(21,29)(22,28)(23,30)(25,31)(27,32)(33,36,37,44)(34,40,38,45)(35,43,41,47)(39,46,42,48)Polycyclic presentation:
2^9|a^2=fh,b^2=g,c^2=fi,d^2=f,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dg,e^b=ef,d^c=df68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4391337);