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 | |
4239 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)(3,4)(5,13)(6,7)(8,9)(10,14)(11,15)(12,16)(17,18,22,21)(19,25,26,23)(20,28,30,24)(27,31,29,32)(33,34,38,40)(35,39,41,36)(37,45,47,42)(43,44,46,48);(2,7)(4,8)(5,14)(11,16)(18,21)(23,25)(24,28)(31,32)(33,35)(34,39)(36,40)(37,46)(38,41)(42,44)(43,47)(45,48);(2,7)(3,9)(10,13)(11,16)(19,26)(23,25)(27,29)(31,32)(34,40)(35,41)(42,45)(43,46);(1,3)(2,8)(4,7)(5,11)(6,9)(10,15)(12,13)(14,16)(17,19,22,26)(18,23,21,25)(20,29,30,27)(24,32,28,31)(33,35)(34,39)(36,40)(37,43)(38,41)(42,48)(44,45)(46,47);(1,4,6,8)(2,9,7,3)(5,12,14,15)(10,16,13,11)(17,20)(18,24)(19,27)(21,28)(22,30)(23,31)(25,32)(26,29)(33,36,38,39)(34,41,40,35)(37,44,47,48)(42,46,45,43);(1,5)(2,10)(3,11)(4,12)(6,14)(7,13)(8,15)(9,16)(17,21,22,18)(19,25,26,23)(20,28,30,24)(27,32,29,31)(33,37,38,47)(34,42,40,45)(35,43,41,46)(36,44,39,48)Polycyclic presentation:
2^9|a^2=gh,d^2=ghi,e^2=h,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^d=egi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8746262);