Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4861280 |
| 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 | |
1376 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 48 points:
(2,5)(6,8)(11,15)(14,16)(17,18,22,20)(19,25,26,23)(21,28,29,24)(27,30,32,31)(33,34)(35,39)(36,38)(37,44)(40,43)(41,45)(42,47)(46,48);(10,13)(14,16)(18,20)(19,26)(24,28)(27,32)(33,35,38,43)(34,39,36,40)(37,42,45,48)(41,46,44,47);(9,10)(11,14)(12,13)(15,16)(17,19,22,26)(18,23,20,25)(21,27,29,32)(24,30,28,31)(33,35)(34,40)(36,39)(37,42)(38,43)(41,47)(44,46)(45,48);(1,2)(3,6)(4,5)(7,8)(17,20,22,18)(19,25,26,23)(21,24,29,28)(27,30,32,31)(33,36,38,34)(35,39,43,40)(37,44,45,41)(42,46,48,47);(1,3,4,7)(2,6,5,8)(9,11,12,15)(10,14,13,16)(17,21,22,29)(18,24,20,28)(19,27,26,32)(23,30,25,31)(33,37,38,45)(34,41,36,44)(35,42,43,48)(39,46,40,47)Polycyclic presentation:
2^9|a^2=f,b^2=fg,c^2=f,d^2=g,e^2=fghi,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dg,e^d=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4815735);