Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4465744 |
| 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 | |
1314 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)(3,7)(4,6)(5,8)(11,15)(14,16)(17,18,21,24)(19,22,25,28)(20,23,27,30)(26,29,31,32)(33,34,37,38)(35,41,43,39)(36,40,44,45)(42,47,48,46);(1,3,6,8)(2,5,4,7)(19,25)(22,28)(26,31)(29,32)(34,38)(36,44)(39,41)(42,48);(2,4)(3,8)(9,10,13,12)(11,14,15,16)(17,18)(19,22)(20,23)(21,24)(25,28)(26,29)(27,30)(31,32)(33,35,37,43)(34,39,38,41)(36,42,44,48)(40,46,45,47);(1,4,6,2)(3,5,8,7)(9,11)(10,14)(12,16)(13,15)(17,19,21,25)(18,22,24,28)(20,26,27,31)(23,29,30,32)(33,34)(35,41)(36,40)(37,38)(39,43)(42,47)(44,45)(46,48);(1,5)(2,3)(4,8)(6,7)(9,12,13,10)(11,16,15,14)(17,20,21,27)(18,23,24,30)(19,26,25,31)(22,29,28,32)(33,36,37,44)(34,40,38,45)(35,42,43,48)(39,46,41,47)Polycyclic presentation:
2^9|a^2=gh,b^2=h,c^2=hi,d^2=fgh,e^2=fghi,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dg,e^b=ef,d^c=df68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4504852);