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 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4158 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,5)(3,9,11,7)(4,12,13,8)(10,14,16,15)(17,18)(19,26)(20,23)(21,28)(22,24)(25,30)(27,31)(29,32)(33,34,39,43)(35,44,38,40)(36,42,37,41)(45,47,46,48);(1,3)(2,7)(4,10)(5,9)(6,11)(8,14)(12,15)(13,16)(17,19,23,22)(18,24,20,26)(21,27,30,29)(25,31,28,32)(33,35,39,38)(34,40,43,44)(36,45,37,46)(41,47,42,48);(17,18,23,20)(19,26,22,24)(21,25,30,28)(27,32,29,31)(33,36,39,37)(34,41,43,42)(35,45,38,46)(40,47,44,48);(1,4,6,13)(2,8,5,12)(3,10,11,16)(7,14,9,15)(17,20,23,18)(19,24,22,26)(21,28,30,25)(27,31,29,32)(33,37,39,36)(34,42,43,41)(35,45,38,46)(40,47,44,48);(1,3)(2,9)(4,10)(5,7)(6,11)(8,15)(12,14)(13,16)(17,21)(18,25)(19,27)(20,28)(22,29)(23,30)(24,31)(26,32)(33,34)(35,40)(36,41)(37,42)(38,44)(39,43)(45,47)(46,48);(1,5,6,2)(3,9,11,7)(4,8,13,12)(10,14,16,15)(17,22,23,19)(18,24,20,26)(21,29,30,27)(25,31,28,32)(33,38,39,35)(34,40,43,44)(36,46,37,45)(41,47,42,48)Polycyclic presentation:
2^9|a^2=gi,b^2=h,c^2=h,d^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^d=fgi,f^e=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8568552);