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 | |
4024 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,9)(3,10,5,7)(4,12,14,8)(11,15,13,16)(18,23)(19,28)(26,30)(27,32)(33,34,39,43)(35,44,45,40)(36,41,47,46)(37,38,42,48);(1,3,6,5)(2,7,9,10)(4,11,14,13)(8,15,12,16)(17,18)(19,24)(20,26)(21,27)(22,23)(25,28)(29,30)(31,32)(35,45)(36,47)(40,44)(41,46);(17,19)(18,24)(20,27)(21,26)(22,28)(23,25)(29,32)(30,31)(33,35,39,45)(34,40,43,44)(36,42,47,37)(38,46,48,41);(1,2)(3,7)(4,8)(5,10)(6,9)(11,15)(12,14)(13,16)(17,19)(18,25)(20,27)(21,30)(22,28)(23,24)(26,31)(29,32)(33,36)(34,41)(35,42)(37,45)(38,40)(39,47)(43,46)(44,48);(1,4)(2,8)(3,11)(5,13)(6,14)(7,15)(9,12)(10,16)(17,20,22,29)(18,26,23,30)(19,27,28,32)(21,25,31,24)(33,37,39,42)(34,38,43,48)(35,36,45,47)(40,46,44,41);(1,5,6,3)(2,7,9,10)(4,13,14,11)(8,15,12,16)(17,21,22,31)(18,27,23,32)(19,26,28,30)(20,25,29,24)(33,38)(34,42)(35,46)(36,44)(37,43)(39,48)(40,47)(41,45)Polycyclic presentation:
2^9|a^2=gh,b^2=ghi,c^2=i,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,f^d=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8344013);