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 | |
4066 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)(3,11)(4,9)(5,15)(6,7)(8,13)(10,16)(12,14)(17,18,23,29)(19,24,31,32)(20,26,30,25)(21,27,28,22)(33,34,39,37)(35,40,45,43)(36,41,47,38)(42,44,46,48);(2,7)(4,14)(8,11)(10,15)(17,19,23,31)(18,24,29,32)(20,28,30,21)(22,25,27,26)(33,35)(34,40)(36,46)(37,43)(38,44)(39,45)(41,48)(42,47);(1,3,6,13)(2,8,7,11)(4,15,14,10)(5,9,16,12)(17,20,23,30)(18,25,29,26)(19,21,31,28)(22,32,27,24)(33,36,39,47)(34,41,37,38)(35,42,45,46)(40,44,43,48);(1,3)(2,8)(4,15)(5,12)(6,13)(7,11)(9,16)(10,14)(17,20)(18,26)(19,28)(21,31)(22,24)(23,30)(25,29)(27,32)(33,37,39,34)(35,43,45,40)(36,41,47,38)(42,44,46,48);(1,4,6,14)(2,9,7,12)(3,10,13,15)(5,11,16,8)(17,21,23,28)(18,27,29,22)(19,30,31,20)(24,25,32,26)(33,38,39,41)(34,36,37,47)(35,44,45,48)(40,46,43,42);(1,5)(2,10)(3,12)(4,11)(6,16)(7,15)(8,14)(9,13)(17,22,23,27)(18,28,29,21)(19,25,31,26)(20,24,30,32)(33,35)(34,40)(36,46)(37,43)(38,44)(39,45)(41,48)(42,47)Polycyclic presentation:
2^9|a^2=i,b^2=gh,c^2=gi,d^2=ghi,e^2=gi,f^2=gh,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, 8922526);