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 | |
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)(3,6)(4,8)(5,7)(9,10,12,14)(11,15,16,13)(20,27)(24,29)(25,31)(30,32)(33,34)(35,38)(36,39)(37,42)(40,46)(41,43)(44,48)(45,47);(1,3,5,8)(2,6,7,4)(18,22)(23,28)(24,29)(30,32)(34,35)(36,43)(40,42)(44,45);(1,4)(2,3)(5,6)(7,8)(17,18)(19,23)(20,24)(21,22)(25,30)(26,28)(27,29)(31,32)(33,35,38,34)(36,41,43,39)(37,42,46,40)(44,48,45,47);(1,3)(2,4)(5,8)(6,7)(9,11,12,16)(10,13,14,15)(17,19,21,26)(18,23,22,28)(20,25,27,31)(24,30,29,32)(33,36)(34,39)(35,41)(37,45)(38,43)(40,48)(42,47)(44,46);(17,20,21,27)(18,24,22,29)(19,25,26,31)(23,30,28,32)(33,37,38,46)(34,40,35,42)(36,44,43,45)(39,47,41,48)Polycyclic presentation:
2^9|a^2=fgi,b^2=fg,c^2=f,d^2=ghi,e^2=h,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, 4803042);