Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4597856 |
| 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 | |
1362 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 48 points:
(2,5)(6,8)(11,15)(13,16)(17,18,21,22)(19,25,26,23)(20,24,29,28)(27,31,32,30)(33,34)(35,40)(36,41)(37,42)(38,39)(43,48)(44,45)(46,47);(9,10,12,14)(11,13,15,16)(18,22)(19,26)(24,28)(27,32)(34,39)(35,41)(42,47)(43,45);(9,10)(11,13)(12,14)(15,16)(17,18)(19,23)(20,28)(21,22)(24,29)(25,26)(27,31)(30,32)(33,35,38,41)(34,40,39,36)(37,45,46,43)(42,44,47,48);(1,2)(3,6)(4,5)(7,8)(17,19,21,26)(18,23,22,25)(20,27,29,32)(24,30,28,31)(33,36,38,40)(34,41,39,35)(37,44,46,48)(42,45,47,43);(1,3,4,7)(2,6,5,8)(9,11,12,15)(10,13,14,16)(17,20)(18,24)(19,27)(21,29)(22,28)(23,30)(25,31)(26,32)(33,37)(34,42)(35,43)(36,44)(38,46)(39,47)(40,48)(41,45)Polycyclic presentation:
2^9|a^2=g,b^2=fh,c^2=fg,d^2=f,e^2=fghi,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^b=dg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4656538);