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 | C22×D4 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3878 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(17,18)(19,25)(20,30)(21,26)(22,24)(23,27)(28,31)(29,32)(33,34,39,42)(35,43,41,37)(36,46,38,40)(44,48,45,47);(1,2)(3,8)(4,5)(6,7)(9,13)(10,14)(11,16)(12,15)(17,19,24,27)(18,25,22,23)(20,29,21,28)(26,31,30,32)(33,35,39,41)(34,37,42,43)(36,44,38,45)(40,48,46,47);(2,7)(4,13)(8,15)(10,16)(17,20,24,21)(18,26,22,30)(19,28,27,29)(23,31,25,32)(33,36,39,38)(34,40,42,46)(35,44,41,45)(37,48,43,47);(1,3)(2,8)(4,10)(5,14)(6,12)(7,15)(9,11)(13,16)(17,21,24,20)(18,26,22,30)(19,29,27,28)(23,31,25,32)(33,34)(35,43)(36,40)(37,41)(38,46)(39,42)(44,47)(45,48);(1,4)(2,9)(3,10)(5,7)(6,13)(8,11)(12,16)(14,15)(17,22,24,18)(19,25,27,23)(20,26,21,30)(28,31,29,32)(33,37,39,43)(34,41,42,35)(36,47,38,48)(40,44,46,45);(1,5,6,9)(2,4,7,13)(3,11,12,14)(8,16,15,10)(17,23,24,25)(18,27,22,19)(20,31,21,32)(26,29,30,28)(33,38,39,36)(34,40,42,46)(35,45,41,44)(37,48,43,47)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=h,d^2=gh,e^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8540627);