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 | |
3772 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,7,9)(3,10,6,8)(4,5,14,15)(11,12,13,16)(17,18)(19,24)(20,28)(21,26)(22,23)(25,30)(27,29)(31,32)(35,41)(36,43)(38,44)(39,46);(1,3,7,6)(2,8,9,10)(4,13,14,11)(5,12,15,16)(18,23)(19,26)(25,28)(27,32)(33,34,37,40)(35,38,41,44)(36,39,43,46)(42,45,47,48);(2,9)(4,14)(8,10)(11,13)(17,19,22,26)(18,24,23,21)(20,27,30,32)(25,31,28,29)(33,35)(34,38)(36,42)(37,41)(39,45)(40,44)(43,47)(46,48);(1,4)(2,5)(3,11)(6,13)(7,14)(8,16)(9,15)(10,12)(17,20,22,30)(18,25,23,28)(19,27,26,32)(21,29,24,31)(33,34)(35,38)(36,39)(37,40)(41,44)(42,45)(43,46)(47,48);(1,5)(2,4)(3,12)(6,16)(7,15)(8,13)(9,14)(10,11)(17,21,22,24)(18,26,23,19)(20,29,30,31)(25,32,28,27)(33,36,37,43)(34,39,40,46)(35,42,41,47)(38,45,44,48);(1,6,7,3)(2,8,9,10)(4,11,14,13)(5,12,15,16)Polycyclic presentation:
2^9|a^2=gi,b^2=h,c^2=i,d^2=i,e^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8658490);