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 | |
4020 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,8)(3,9,11,5)(4,7,13,14)(10,15,16,12)(18,23)(20,24)(26,29)(28,31)(33,34,38,39)(35,37,42,40)(36,44,46,41)(43,47,48,45);(1,3,6,11)(2,5,8,9)(4,10,13,16)(7,12,14,15)(17,18)(19,24)(20,25)(21,26)(22,23)(27,31)(28,32)(29,30)(34,39)(35,42)(41,44)(43,48);(1,2)(3,5)(4,7)(6,8)(9,11)(10,12)(13,14)(15,16)(17,19,22,25)(18,24,23,20)(21,27,30,32)(26,31,29,28)(33,35,38,42)(34,40,39,37)(36,43,46,48)(41,47,44,45);(1,2)(3,9)(4,7)(5,11)(6,8)(10,15)(12,16)(13,14)(17,18)(19,24)(20,25)(21,29)(22,23)(26,30)(27,28)(31,32)(33,34)(35,40)(36,41)(37,42)(38,39)(43,47)(44,46)(45,48);(1,4,6,13)(2,7,8,14)(3,10,11,16)(5,12,9,15)(17,20,22,24)(18,25,23,19)(21,28,30,31)(26,32,29,27)(33,36,38,46)(34,41,39,44)(35,43,42,48)(37,45,40,47);(1,5,6,9)(2,3,8,11)(4,12,13,15)(7,10,14,16)(17,21,22,30)(18,26,23,29)(19,27,25,32)(20,28,24,31)(33,37,38,40)(34,42,39,35)(36,45,46,47)(41,48,44,43)Polycyclic presentation:
2^9|a^2=h,b^2=hi,c^2=ghi,e^2=g,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, 8343421);