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 | |
3829 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,7,11)(3,8,6,10)(4,9,15,5)(12,14,13,16)(17,18)(19,27)(20,21)(22,24)(23,26)(25,31)(28,29)(30,32)(33,34,39,36)(35,42,38,40)(37,44,46,41)(43,47,45,48);(1,3,7,6)(2,8,11,10)(4,13,15,12)(5,14,9,16)(17,19,23,22)(18,24,26,27)(20,30,25,28)(21,29,31,32)(33,35,39,38)(34,40,36,42)(37,43,46,45)(41,47,44,48);(1,4,7,15)(2,9,11,5)(3,12,6,13)(8,14,10,16)(17,19,23,22)(18,24,26,27)(20,30,25,28)(21,29,31,32)(33,36,39,34)(35,40,38,42)(37,44,46,41)(43,47,45,48);(1,4)(2,5)(3,13)(6,12)(7,15)(8,14)(9,11)(10,16)(17,20,23,25)(18,21,26,31)(19,28,22,30)(24,32,27,29)(33,37)(34,41)(35,43)(36,44)(38,45)(39,46)(40,47)(42,48);(1,5)(2,4)(3,14)(6,16)(7,9)(8,13)(10,12)(11,15)(17,21)(18,25)(19,29)(20,26)(22,32)(23,31)(24,28)(27,30)(33,38,39,35)(34,40,36,42)(37,45,46,43)(41,47,44,48);(1,6,7,3)(2,10,11,8)(4,13,15,12)(5,14,9,16)(17,22,23,19)(18,24,26,27)(20,28,25,30)(21,29,31,32)Polycyclic presentation:
2^9|a^2=i,b^2=hi,c^2=hi,d^2=h,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, 8657466);