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, 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:
(1,2,4,6)(3,5,7,8)(9,10,13,12)(11,14,16,15)(17,18,21,24)(19,25,26,22)(20,28,29,23)(27,30,32,31)(33,34)(35,41)(36,40)(37,38)(39,43)(42,48)(44,46)(45,47);(1,3,4,7)(2,5,6,8)(9,11,13,16)(10,14,12,15)(19,26)(20,29)(22,25)(23,28)(34,38)(36,44)(39,41)(42,45);(1,3)(2,5)(4,7)(6,8)(9,12,13,10)(11,14,16,15)(17,18)(19,25)(20,23)(21,24)(22,26)(27,31)(28,29)(30,32)(33,35)(34,39)(36,45)(37,43)(38,41)(40,48)(42,44)(46,47);(9,11)(10,15)(12,14)(13,16)(17,19)(18,22)(20,27)(21,26)(23,30)(24,25)(28,31)(29,32)(33,34)(35,39)(36,40)(37,38)(41,43)(42,47)(44,46)(45,48);(1,2)(3,5)(4,6)(7,8)(17,20)(18,23)(19,27)(21,29)(22,30)(24,28)(25,31)(26,32)(33,36,37,44)(34,40,38,46)(35,42,43,45)(39,47,41,48)Polycyclic presentation:
2^9|a^2=fgh,b^2=h,c^2=fgi,e^2=f,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, 6143473);