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 | |
3906 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)(3,10)(4,8)(5,6)(7,9)(11,16)(12,13)(14,15)(17,18,22,21)(19,23,25,27)(20,24,28,29)(26,30,31,32)(33,34,40,44)(35,45,48,41)(36,42,37,39)(38,43,47,46);(1,3,7,4)(2,8,9,10)(5,14,11,12)(6,13,16,15)(17,19)(18,23)(20,26)(21,27)(22,25)(24,30)(28,31)(29,32)(33,35)(34,41)(36,43)(37,46)(38,42)(39,47)(40,48)(44,45);(2,9)(6,16)(8,10)(13,15)(19,25)(23,27)(26,31)(30,32)(33,36,40,37)(34,42,44,39)(35,43,48,46)(38,45,47,41);(1,4,7,3)(2,10,9,8)(5,12,11,14)(6,15,16,13)(17,20)(18,24)(19,26)(21,29)(22,28)(23,30)(25,31)(27,32)(33,37,40,36)(34,42,44,39)(35,46,48,43)(38,45,47,41);(1,5,7,11)(2,6,9,16)(3,12,4,14)(8,15,10,13)(17,21,22,18)(19,27,25,23)(20,24,28,29)(26,30,31,32)(33,38,40,47)(34,43,44,46)(35,42,48,39)(36,45,37,41);(1,6)(2,11)(3,13)(4,15)(5,9)(7,16)(8,12)(10,14)(17,19)(18,23)(20,26)(21,27)(22,25)(24,30)(28,31)(29,32)(33,39,40,42)(34,36,44,37)(35,47,48,38)(41,43,45,46)Polycyclic presentation:
2^9|a^2=ghi,b^2=h,c^2=gh,d^2=g,e^2=gi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,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, 8807243);