Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5605735 |
| 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 | |
1302 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)(3,6)(4,5)(7,8)(9,10,13,11)(12,15,16,14)(17,18,22,23)(19,20,25,24)(21,26,29,31)(27,28,32,30)(35,41)(36,43)(38,44)(39,46);(1,3)(2,6)(4,7)(5,8)(18,23)(19,25)(21,29)(28,30)(33,34,37,40)(35,38,41,44)(36,39,43,46)(42,45,47,48);(1,3,5,8)(2,7,4,6)(9,11,13,10)(12,15,16,14)(17,19,22,25)(18,24,23,20)(21,27,29,32)(26,30,31,28)(33,34)(35,38)(36,39)(37,40)(41,44)(42,45)(43,46)(47,48);(1,3)(2,7)(4,6)(5,8)(9,12)(10,14)(11,15)(13,16)(17,20,22,24)(18,25,23,19)(21,30,29,28)(26,27,31,32)(33,35,37,41)(34,38,40,44)(36,42,43,47)(39,45,46,48);(1,4,5,2)(3,6,8,7)(17,21)(18,26)(19,27)(20,28)(22,29)(23,31)(24,30)(25,32)(33,36)(34,39)(35,42)(37,43)(38,45)(40,46)(41,47)(44,48)Polycyclic presentation:
2^9|a^2=hi,b^2=gh,c^2=fghi,d^2=fgh,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=eg,d^c=dfg,e^d=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5638527);