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 | |
4216 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)(3,8)(4,5)(6,14)(7,9)(10,15)(11,13)(12,16)(17,18,22,27)(19,25,30,23)(20,21,26,24)(28,32,31,29)(33,34)(35,41)(36,44)(37,43)(38,39)(40,45)(42,47)(46,48);(1,3,7,4)(2,8,9,5)(6,11,15,12)(10,16,14,13)(17,19)(18,23)(20,28)(21,29)(22,30)(24,32)(25,27)(26,31)(33,35,38,43)(34,37,39,41)(36,42,45,46)(40,47,44,48);(2,9)(3,4)(6,15)(13,16)(20,26)(21,24)(28,31)(29,32)(34,39)(36,45)(37,41)(42,46);(1,4,7,3)(2,5,9,8)(6,12,15,11)(10,13,14,16)(17,20,22,26)(18,24,27,21)(19,28,30,31)(23,32,25,29)(33,36)(34,40)(35,42)(37,47)(38,45)(39,44)(41,48)(43,46);(1,5)(2,4)(3,9)(6,13)(7,8)(10,12)(11,14)(15,16)(17,19)(18,25)(20,31)(21,29)(22,30)(23,27)(24,32)(26,28)(33,35)(34,41)(36,46)(37,39)(38,43)(40,47)(42,45)(44,48);(1,6)(2,10)(3,11)(4,12)(5,13)(7,15)(8,16)(9,14)(17,21,22,24)(18,26,27,20)(19,29,30,32)(23,31,25,28)(33,37,38,41)(34,35,39,43)(36,47,45,48)(40,42,44,46)Polycyclic presentation:
2^9|a^2=ghi,b^2=h,d^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8812361);