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 | C2×#121 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3866 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)(3,11)(4,13)(5,10)(6,14)(7,9)(8,12)(15,16)(17,18,23,21)(19,26,20,24)(22,25,30,29)(27,32,28,31)(33,34,39,37)(35,40,43,36)(38,45,47,41)(42,44,46,48);(1,3,7,12)(2,8,9,11)(4,5,14,15)(6,10,13,16)(19,20)(24,26)(27,28)(31,32)(33,35,39,43)(34,40,37,36)(38,42,47,46)(41,48,45,44);(2,9)(3,12)(4,14)(10,16)(17,19,23,20)(18,24,21,26)(22,27,30,28)(25,31,29,32)(33,36,39,40)(34,35,37,43)(38,44,47,48)(41,42,45,46);(1,4)(2,6)(3,5)(7,14)(8,10)(9,13)(11,16)(12,15)(17,20,23,19)(18,24,21,26)(22,28,30,27)(25,31,29,32)(33,37,39,34)(35,40,43,36)(38,45,47,41)(42,48,46,44);(1,5)(2,10)(3,4)(6,8)(7,15)(9,16)(11,13)(12,14)(17,21,23,18)(19,26,20,24)(22,25,30,29)(27,31,28,32)(33,35)(34,40)(36,37)(38,46)(39,43)(41,44)(42,47)(45,48);(1,6)(2,4)(3,10)(5,8)(7,13)(9,14)(11,15)(12,16)(17,22)(18,25)(19,27)(20,28)(21,29)(23,30)(24,31)(26,32)(33,38)(34,41)(35,42)(36,44)(37,45)(39,47)(40,48)(43,46)Polycyclic presentation:
2^9|a^2=gi,b^2=ghi,c^2=gi,d^2=gi,e^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fghi,c^b=ci,d^b=dgi,e^b=eghi,d^c=dghi,e^d=egi,f^e=fgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10227512);