Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4379698 |
| 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 | |
1362 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,6,5)(3,8,7,4)(9,10,13,16)(11,14,12,15)(19,25)(20,27)(23,28)(24,29)(33,34,37,38)(35,39,42,45)(36,40,44,46)(41,47,43,48);(9,11,13,12)(10,14,16,15)(18,22)(23,28)(24,29)(30,32)(34,38)(39,45)(40,46)(47,48);(1,3)(2,4)(5,8)(6,7)(9,10)(11,14)(12,15)(13,16)(17,18,21,22)(19,23,25,28)(20,24,27,29)(26,30,31,32)(33,35,37,42)(34,39,38,45)(36,43,44,41)(40,48,46,47);(1,4,6,8)(2,7,5,3)(9,11,13,12)(10,14,16,15)(17,19,21,25)(18,23,22,28)(20,26,27,31)(24,30,29,32)(33,34)(35,39)(36,40)(37,38)(41,47)(42,45)(43,48)(44,46);(1,5,6,2)(3,8,7,4)(9,12,13,11)(10,15,16,14)(17,20)(18,24)(19,26)(21,27)(22,29)(23,30)(25,31)(28,32)(33,36,37,44)(34,40,38,46)(35,41,42,43)(39,47,45,48)Polycyclic presentation:
2^9|a^2=fg,b^2=fghi,c^2=fh,d^2=gh,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,e^c=ef80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4292437);