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 | |
4033 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)(3,9)(4,6)(5,8)(7,11)(10,16)(12,14)(13,15)(33,34,37,40)(35,38,42,46)(36,39,44,43)(41,45,48,47);(1,3,6,11)(2,7,4,9)(5,10,14,13)(8,15,12,16)(17,18,22,23)(19,24,28,31)(20,25,29,21)(26,27,32,30)(33,34)(35,38)(36,39)(37,40)(41,45)(42,46)(43,44)(47,48);(2,4)(7,9)(8,12)(15,16)(18,23)(20,29)(24,31)(26,32)(33,35,37,42)(34,38,40,46)(36,41,44,48)(39,45,43,47);(1,3)(2,7)(4,9)(5,13)(6,11)(8,16)(10,14)(12,15)(17,19)(18,24)(20,26)(21,30)(22,28)(23,31)(25,27)(29,32)(33,34)(35,38)(36,43)(37,40)(39,44)(41,47)(42,46)(45,48);(1,4,6,2)(3,9,11,7)(5,12,14,8)(10,16,13,15)(17,20)(18,25)(19,26)(21,23)(22,29)(24,27)(28,32)(30,31)(33,35)(34,38)(36,41)(37,42)(39,45)(40,46)(43,47)(44,48);(1,5)(2,8)(3,10)(4,12)(6,14)(7,15)(9,16)(11,13)(17,21,22,25)(18,20,23,29)(19,27,28,30)(24,32,31,26)(33,36,37,44)(34,39,40,43)(35,41,42,48)(38,45,46,47)Polycyclic presentation:
2^9|a^2=gh,b^2=hi,c^2=gh,e^2=h,f^2=ghi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^c=egi,f^d=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8456762);