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 | C24 | C25 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3775 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(17,18,23,29)(19,24,31,32)(20,26,30,25)(21,27,28,22)(33,34,40,44)(35,45,47,38)(36,46,48,41)(37,43,39,42);(1,2,6,7)(3,8,11,16)(4,9,14,13)(5,10,15,12)(17,19,23,31)(18,24,29,32)(20,28,30,21)(22,25,27,26)(33,35,40,47)(34,38,44,45)(36,43,48,42)(37,41,39,46);(2,7)(3,11)(4,14)(5,15)(17,20,23,30)(18,25,29,26)(19,21,31,28)(22,32,27,24)(33,36,40,48)(34,41,44,46)(35,42,47,43)(37,45,39,38);(1,3)(2,8)(4,10)(5,13)(6,11)(7,16)(9,15)(12,14)(17,20)(18,26)(19,28)(21,31)(22,24)(23,30)(25,29)(27,32)(33,37,40,39)(34,42,44,43)(35,41,47,46)(36,45,48,38);(1,4)(2,9)(3,12)(5,8)(6,14)(7,13)(10,11)(15,16)(17,21,23,28)(18,27,29,22)(19,30,31,20)(24,25,32,26)(33,38)(34,35)(36,37)(39,48)(40,45)(41,42)(43,46)(44,47);(1,5)(2,10)(3,13)(4,8)(6,15)(7,12)(9,11)(14,16)(17,22,23,27)(18,28,29,21)(19,25,31,26)(20,24,30,32)(33,39,40,37)(34,43,44,42)(35,46,47,41)(36,38,48,45)Polycyclic presentation:
2^9|a^2=h,b^2=i,c^2=h,d^2=g,e^2=gh,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=egi,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, 8886568);