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 | |
4266 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2)(3,5)(4,8)(6,7)(9,13)(10,12)(11,14)(15,16)(17,18,23,25)(19,22,20,24)(21,26,30,32)(27,29,28,31)(33,34)(35,42)(36,39)(37,40)(38,44)(41,46)(43,47)(45,48);(4,11)(8,14)(10,15)(12,16)(17,19,23,20)(18,24,25,22)(21,28,30,27)(26,29,32,31)(35,37)(40,42)(43,45)(47,48);(2,7)(3,9)(8,14)(10,15)(18,25)(19,20)(26,32)(27,28)(33,35,39,37)(34,40,36,42)(38,43,46,45)(41,47,44,48);(1,3,6,9)(2,5,7,13)(4,10,11,15)(8,12,14,16)(17,20,23,19)(18,22,25,24)(21,28,30,27)(26,29,32,31)(33,36,39,34)(35,40,37,42)(38,44,46,41)(43,47,45,48);(1,4,6,11)(2,8,7,14)(3,10,9,15)(5,12,13,16)(17,21)(18,26)(19,27)(20,28)(22,29)(23,30)(24,31)(25,32)(33,37,39,35)(34,40,36,42)(38,43,46,45)(41,48,44,47);(1,5)(2,9)(3,7)(4,12)(6,13)(8,15)(10,14)(11,16)(17,22)(18,19)(20,25)(21,29)(23,24)(26,27)(28,32)(30,31)(33,38)(34,41)(35,43)(36,44)(37,45)(39,46)(40,47)(42,48)Polycyclic presentation:
2^9|a^2=g,b^2=g,c^2=gi,d^2=h,e^2=gh,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dh,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9620665);