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 | |
3752 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)(3,13)(4,10)(5,6)(7,12)(8,15)(9,16)(11,14)(18,22)(20,27)(23,28)(26,31)(33,34,38,41)(35,36,43,39)(37,45,46,40)(42,47,48,44);(1,3)(2,8)(4,5)(6,9)(7,15)(10,11)(12,13)(14,16)(19,25)(23,28)(26,31)(30,32)(33,35,38,43)(34,39,41,36)(37,42,46,48)(40,47,45,44);(1,4,7,16)(2,9,12,10)(3,5,15,14)(6,13,11,8)(17,18)(19,23)(20,24)(21,22)(25,28)(26,30)(27,29)(31,32)(33,36,38,39)(34,35,41,43)(37,44,46,47)(40,42,45,48);(1,4)(2,10)(3,14)(5,15)(6,8)(7,16)(9,12)(11,13)(17,19,21,25)(18,23,22,28)(20,26,27,31)(24,30,29,32)(33,35)(34,39)(36,41)(37,42)(38,43)(40,47)(44,45)(46,48);(1,5,7,14)(2,11,12,6)(3,4,15,16)(8,10,13,9)(17,20,21,27)(18,24,22,29)(19,26,25,31)(23,30,28,32)(33,37,38,46)(34,40,41,45)(35,42,43,48)(36,44,39,47);(1,6,7,11)(2,5,12,14)(3,9,15,10)(4,13,16,8)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=g,d^2=gh,e^2=h,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, 8655110);