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 | |
1363 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)(3,7)(4,8)(5,6)(9,10,13,14)(11,12,15,16)(17,18,22,20)(19,23,25,27)(21,28,30,24)(26,32,29,31)(36,43)(37,45)(39,41)(40,42);(1,3)(2,7)(4,6)(5,8)(11,15)(12,16)(18,20)(19,25)(24,28)(26,29)(33,34,38,35)(36,41,43,39)(37,40,45,42)(44,47,48,46);(1,2,6,5)(3,7,4,8)(9,11)(10,12)(13,15)(14,16)(17,19)(18,23)(20,27)(21,29)(22,25)(24,32)(26,30)(28,31)(33,35,38,34)(36,41,43,39)(37,42,45,40)(44,47,48,46);(1,4,6,3)(2,7,5,8)(9,12)(10,11)(13,16)(14,15)(17,20,22,18)(19,23,25,27)(21,28,30,24)(26,31,29,32)(33,36)(34,39)(35,41)(37,44)(38,43)(40,46)(42,47)(45,48);(1,5,6,2)(3,7,4,8)(17,21)(18,24)(19,26)(20,28)(22,30)(23,31)(25,29)(27,32)(33,37)(34,40)(35,42)(36,44)(38,45)(39,46)(41,47)(43,48)Polycyclic presentation:
2^9|a^2=h,b^2=fghi,c^2=fhi,d^2=fg,e^2=g,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, 6178489);