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, cosocle | C24 | C25 |
| prime radical | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1367 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(3,7)(6,8)(9,10,14,12)(11,15,13,16)(18,22)(20,27)(23,28)(26,31)(33,34)(35,41)(36,44)(37,38)(39,43)(40,46)(42,47)(45,48);(1,2,5,4)(3,6,7,8)(9,11,14,13)(10,15,12,16)(18,22)(19,25)(24,29)(26,31)(34,38)(39,41)(40,44)(47,48);(9,12,14,10)(11,15,13,16)(17,18)(19,23)(20,24)(21,22)(25,28)(26,30)(27,29)(31,32)(33,34)(35,39)(36,40)(37,38)(41,43)(42,47)(44,46)(45,48);(1,3)(2,6)(4,8)(5,7)(9,11,14,13)(10,16,12,15)(17,19,21,25)(18,23,22,28)(20,26,27,31)(24,30,29,32)(33,35,37,43)(34,39,38,41)(36,45,46,42)(40,48,44,47);(1,4,5,2)(3,8,7,6)(9,13,14,11)(10,15,12,16)(17,20)(18,24)(19,26)(21,27)(22,29)(23,30)(25,31)(28,32)(33,36,37,46)(34,40,38,44)(35,42,43,45)(39,47,41,48)Polycyclic presentation:
2^9|a^2=fgh,b^2=fi,c^2=fgh,d^2=h,e^2=i,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, 4827051);