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 | |
1388 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)(9,10,12,14)(11,13,16,15)(17,18,22,20)(19,23,25,26)(21,28,29,24)(27,32,31,30)(35,41)(36,44)(38,42)(39,46);(9,10)(11,13)(12,14)(15,16)(19,25)(23,26)(27,31)(30,32)(33,34,37,40)(35,42,41,38)(36,39,44,46)(43,48,47,45);(17,19,22,25)(18,23,20,26)(21,27,29,31)(24,30,28,32)(33,35,37,41)(34,38,40,42)(36,43,44,47)(39,45,46,48);(1,2)(3,6)(4,5)(7,8)(9,10)(11,15)(12,14)(13,16)(17,20,22,18)(19,26,25,23)(21,28,29,24)(27,32,31,30)(33,34)(35,38)(36,39)(37,40)(41,42)(43,45)(44,46)(47,48);(1,3,4,7)(2,6,5,8)(9,11,12,16)(10,13,14,15)(17,21,22,29)(18,24,20,28)(19,27,25,31)(23,30,26,32)(33,36)(34,39)(35,43)(37,44)(38,45)(40,46)(41,47)(42,48)Polycyclic presentation:
2^9|a^2=fgh,b^2=g,c^2=h,d^2=gh,e^2=ghi,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, 4819461);