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 | |
4236 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,9)(3,7,10,5)(4,8,12,16)(11,15,13,14)(17,18,22,23)(19,25,27,21)(20,28,29,24)(26,31,32,30)(33,34,37,41)(35,39,43,38)(36,40,45,44)(42,47,48,46);(1,3)(2,7)(4,11)(5,9)(6,10)(8,15)(12,13)(14,16)(17,19,22,27)(18,21,23,25)(20,26,29,32)(24,31,28,30)(33,35)(34,38)(36,42)(37,43)(39,41)(40,46)(44,47)(45,48);(3,10)(4,12)(5,7)(8,16)(18,23)(20,29)(21,25)(26,32)(33,34,37,41)(35,38,43,39)(36,44,45,40)(42,47,48,46);(1,3)(2,7)(4,13)(5,9)(6,10)(8,14)(11,12)(15,16)(17,20)(18,24)(19,26)(21,31)(22,29)(23,28)(25,30)(27,32)(33,34)(35,38)(36,40)(37,41)(39,43)(42,46)(44,45)(47,48);(1,4)(2,8)(3,11)(5,14)(6,12)(7,15)(9,16)(10,13)(17,21)(18,19)(20,30)(22,25)(23,27)(24,32)(26,28)(29,31)(33,35)(34,39)(36,42)(37,43)(38,41)(40,47)(44,46)(45,48);(1,5,6,7)(2,3,9,10)(4,14,12,15)(8,11,16,13)(17,18)(19,21)(20,24)(22,23)(25,27)(26,31)(28,29)(30,32)(33,36)(34,40)(35,42)(37,45)(38,46)(39,47)(41,44)(43,48)Polycyclic presentation:
2^9|a^2=gi,b^2=h,c^2=gh,f^2=i,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=egi,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, 8927269);