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 | |
3873 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)(19,24)(20,26)(21,22)(23,28)(25,29)(27,32)(30,31)(33,34,39,44)(35,45,47,40)(36,43,38,41)(37,42,46,48);(1,2,6,7)(3,8,11,16)(4,9,14,13)(5,10,15,12)(17,19,23,30)(18,24,28,31)(20,27,29,21)(22,26,32,25)(33,34)(35,45)(36,41)(37,48)(38,43)(39,44)(40,47)(42,46);(2,7)(3,11)(4,14)(5,15)(17,20,23,29)(18,25,28,26)(19,21,30,27)(22,24,32,31)(33,35,39,47)(34,40,44,45)(36,46,38,37)(41,48,43,42);(1,3)(2,8)(4,10)(5,13)(6,11)(7,16)(9,15)(12,14)(17,20)(18,26)(19,27)(21,30)(22,31)(23,29)(24,32)(25,28)(33,36,39,38)(34,41,44,43)(35,46,47,37)(40,48,45,42);(1,4)(2,9)(3,12)(5,8)(6,14)(7,13)(10,11)(15,16)(17,21,23,27)(18,22,28,32)(19,29,30,20)(24,25,31,26)(33,37)(34,42)(35,36)(38,47)(39,46)(40,41)(43,45)(44,48);(1,5)(2,10)(3,13)(4,8)(6,15)(7,12)(9,11)(14,16)(17,22)(18,27)(19,26)(20,31)(21,28)(23,32)(24,29)(25,30)(33,38,39,36)(34,43,44,41)(35,37,47,46)(40,42,45,48)Polycyclic presentation:
2^9|a^2=g,b^2=gi,c^2=h,d^2=g,e^2=gh,f^2=g,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, 8884491);