Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #6132643 |
| 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 | |
1294 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,6,4)(3,8,5,7)(11,15)(14,16)(17,18,21,22)(19,25,27,23)(20,24,28,29)(26,31,32,30)(33,34,37,40)(35,41,42,38)(36,44,45,39)(43,46,48,47);(1,2,6,4)(3,7,5,8)(10,13)(11,15)(18,22)(20,28)(23,25)(26,32)(35,42)(36,45)(38,41)(39,44);(1,3,6,5)(2,7,4,8)(9,10)(11,14)(12,13)(15,16)(17,18)(19,23)(20,29)(21,22)(24,28)(25,27)(26,31)(30,32)(33,34)(35,41)(36,39)(37,40)(38,42)(43,47)(44,45)(46,48);(1,4,6,2)(3,7,5,8)(9,11)(10,14)(12,15)(13,16)(17,19,21,27)(18,23,22,25)(20,26,28,32)(24,30,29,31)(33,35,37,42)(34,38,40,41)(36,43,45,48)(39,46,44,47);(1,5,6,3)(2,7,4,8)(17,20)(18,24)(19,26)(21,28)(22,29)(23,30)(25,31)(27,32)(33,36,37,45)(34,39,40,44)(35,43,42,48)(38,46,41,47)Polycyclic presentation:
2^9|a^2=g,b^2=fghi,c^2=fghi,d^2=g,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=ehi,c^b=ch,d^b=dfi,e^b=eg,d^c=dfg,e^c=ef62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6178017);