Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C26 |
| prime radical | C22×D4 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3909 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,7,10)(3,8,5,4)(6,9,15,14)(11,16,13,12)(17,18,22,23)(19,21,26,24)(20,28,30,25)(27,31,32,29)(33,34)(35,36)(37,41)(38,39)(40,44)(42,43)(45,46)(47,48);(1,3,7,5)(2,4,10,8)(6,11,15,13)(9,12,14,16)(18,23)(19,26)(25,28)(27,32)(34,39)(35,40)(41,46)(42,47);(1,4,7,8)(2,3,10,5)(6,12,15,16)(9,11,14,13)(17,18)(19,24)(20,25)(21,26)(22,23)(27,31)(28,30)(29,32)(33,35,38,40)(34,36,39,44)(37,42,45,47)(41,43,46,48);(1,2)(3,4)(5,8)(6,14)(7,10)(9,15)(11,16)(12,13)(17,19,22,26)(18,24,23,21)(20,27,30,32)(25,31,28,29)(33,36)(34,40)(35,39)(37,43)(38,44)(41,47)(42,46)(45,48);(1,5,7,3)(2,8,10,4)(6,13,15,11)(9,16,14,12)(17,20,22,30)(18,25,23,28)(19,27,26,32)(21,29,24,31);(1,6)(2,9)(3,11)(4,12)(5,13)(7,15)(8,16)(10,14)(17,21,22,24)(18,26,23,19)(20,29,30,31)(25,32,28,27)(33,37,38,45)(34,41,39,46)(35,42,40,47)(36,43,44,48)Polycyclic presentation:
2^9|a^2=ghi,b^2=gi,c^2=gh,d^2=h,e^2=ghi,f^2=i,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,f^d=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8370392);