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 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4056 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,7,12)(3,10,5,8)(4,14,13,9)(6,11,15,16)(18,21)(23,25)(24,29)(30,31)(33,34,39,36)(35,42,41,38)(37,40,45,44)(43,48,47,46);(1,3,7,5)(2,8,12,10)(4,15,13,6)(9,16,14,11)(18,21)(20,28)(23,25)(26,32)(37,45)(40,44)(43,47)(46,48);(1,4)(2,9)(3,6)(5,15)(7,13)(8,11)(10,16)(12,14)(17,18,22,21)(19,25,27,23)(20,29,28,24)(26,30,32,31)(33,35)(34,38)(36,42)(37,43)(39,41)(40,46)(44,48)(45,47);(1,5,7,3)(2,10,12,8)(4,15,13,6)(9,16,14,11)(17,19,22,27)(18,23,21,25)(20,26,28,32)(24,30,29,31)(33,36,39,34)(35,42,41,38)(37,44,45,40)(43,48,47,46);(1,6,7,15)(2,11,12,16)(3,13,5,4)(8,14,10,9)(17,20,22,28)(18,24,21,29)(19,26,27,32)(23,30,25,31)(33,37,39,45)(34,40,36,44)(35,43,41,47)(38,46,42,48);(1,2)(3,8)(4,9)(5,10)(6,11)(7,12)(13,14)(15,16)(17,21,22,18)(19,25,27,23)(20,29,28,24)(26,31,32,30)(33,38,39,42)(34,41,36,35)(37,46,45,48)(40,47,44,43)Polycyclic presentation:
2^9|a^2=ghi,b^2=g,c^2=gi,d^2=h,e^2=h,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8679884);