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 |
| prime radical | #2456 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3944 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,10)(3,7,13,15)(4,8,14,12)(5,9,16,11)(17,18,24,22)(19,28,31,25)(20,26,21,30)(23,27,32,29)(35,43)(37,48)(39,44)(41,45);(1,3)(2,7)(4,9)(5,12)(6,13)(8,16)(10,15)(11,14)(17,19)(18,25)(20,29)(21,27)(22,28)(23,30)(24,31)(26,32)(33,34,38,42)(35,44,43,39)(36,40,47,46)(37,41,48,45);(1,4)(2,8)(3,11)(5,7)(6,14)(9,13)(10,12)(15,16)(17,20,24,21)(18,26,22,30)(19,29,31,27)(23,25,32,28)(33,35,38,43)(34,39,42,44)(36,45,47,41)(37,40,48,46);(1,4)(2,8)(3,9)(5,15)(6,14)(7,16)(10,12)(11,13)(17,21,24,20)(18,26,22,30)(19,27,31,29)(23,25,32,28)(33,36)(34,40)(35,45)(37,44)(38,47)(39,48)(41,43)(42,46);(1,3)(2,7)(4,11)(5,8)(6,13)(9,14)(10,15)(12,16)(17,22,24,18)(19,28,31,25)(20,30,21,26)(23,27,32,29)(33,34)(35,39)(36,46)(37,45)(38,42)(40,47)(41,48)(43,44);(1,5)(2,9)(3,12)(4,7)(6,16)(8,13)(10,11)(14,15)(17,23,24,32)(18,27,22,29)(19,30,31,26)(20,25,21,28)(33,37)(34,41)(35,40)(36,39)(38,48)(42,45)(43,46)(44,47)Polycyclic presentation:
2^9|a^2=ghi,b^2=h,c^2=gh,d^2=g,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,e^d=egi,f^d=fgi,f^e=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9121934);