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, cosocle | C24 | C25 |
| prime radical | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1486 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)(3,6)(4,8)(5,7)(10,14)(12,16)(17,18,21,24)(19,25,26,22)(20,23,28,30)(27,31,32,29)(33,34,37,38)(35,41,42,39)(36,40,45,46)(43,48,44,47);(1,3,5,8)(2,6,7,4)(10,14)(11,15)(19,26)(22,25)(27,32)(29,31)(34,38)(35,42)(40,46)(43,44);(1,3)(2,4)(5,8)(6,7)(9,10,13,14)(11,12,15,16)(17,18)(19,22)(20,23)(21,24)(25,26)(27,29)(28,30)(31,32)(33,35,37,42)(34,39,38,41)(36,44,45,43)(40,48,46,47);(1,3)(2,4)(5,8)(6,7)(9,11,13,15)(10,12,14,16)(17,19,21,26)(18,22,24,25)(20,27,28,32)(23,29,30,31)(33,34)(35,39)(36,40)(37,38)(41,42)(43,47)(44,48)(45,46);(1,4)(2,3)(5,6)(7,8)(9,12)(10,15)(11,14)(13,16)(17,20,21,28)(18,23,24,30)(19,27,26,32)(22,29,25,31)(33,36,37,45)(34,40,38,46)(35,43,42,44)(39,47,41,48)Polycyclic presentation:
2^9|a^2=fgh,b^2=fghi,c^2=fgi,d^2=hi,e^2=fgh,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, 4637139);