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 | |
3851 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(17,18,23,26)(19,21,20,24)(22,25,30,32)(27,29,28,31)(33,34,37,38)(35,41,42,39)(36,44,45,40)(43,46,48,47);(1,2)(3,8)(4,5)(6,7)(9,14)(10,16)(11,13)(12,15)(17,19,23,20)(18,24,26,21)(22,27,30,28)(25,31,32,29)(34,38)(35,42)(40,44)(43,48);(2,7)(5,9)(8,15)(11,16)(17,19,23,20)(18,21,26,24)(22,28,30,27)(25,31,32,29)(33,35,37,42)(34,39,38,41)(36,43,45,48)(40,46,44,47);(1,3)(2,8)(4,13)(5,11)(6,12)(7,15)(9,16)(10,14)(17,20,23,19)(18,21,26,24)(22,28,30,27)(25,29,32,31)(33,34)(35,39)(36,40)(37,38)(41,42)(43,46)(44,45)(47,48);(1,4)(2,9)(3,10)(5,7)(6,14)(8,11)(12,13)(15,16)(17,21)(18,20)(19,26)(22,29)(23,24)(25,28)(27,32)(30,31)(33,36,37,45)(34,40,38,44)(35,43,42,48)(39,46,41,47);(1,5,6,9)(2,4,7,14)(3,11,12,16)(8,13,15,10)(17,22,23,30)(18,25,26,32)(19,27,20,28)(21,29,24,31)Polycyclic presentation:
2^9|a^2=g,b^2=gh,c^2=g,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,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, 8279342);