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 | |
4003 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)(3,9)(4,8)(5,10)(6,7)(11,16)(12,13)(14,15)(17,18)(19,27)(20,23)(21,25)(22,30)(24,28)(26,31)(29,32)(33,34,40,44)(35,45,48,41)(36,42,37,47)(38,43,46,39);(1,3)(2,5)(4,12)(6,10)(7,9)(8,16)(11,14)(13,15)(17,19,23,28)(18,24,20,27)(21,31,32,22)(25,30,29,26)(33,35)(34,41)(36,46)(37,38)(39,47)(40,48)(42,43)(44,45);(2,7)(3,10)(8,15)(11,12)(18,20)(24,27)(25,29)(26,30)(33,36,40,37)(34,42,44,47)(35,46,48,38)(39,41,43,45);(1,3)(2,5)(4,11)(6,10)(7,9)(8,13)(12,14)(15,16)(17,20,23,18)(19,24,28,27)(21,25,32,29)(22,30,31,26)(33,37,40,36)(34,42,44,47)(35,46,48,38)(39,45,43,41);(1,4,6,14)(2,8,7,15)(3,11,10,12)(5,13,9,16)(17,21,23,32)(18,25,20,29)(19,22,28,31)(24,26,27,30)(33,38,40,46)(34,43,44,39)(35,37,48,36)(41,42,45,47);(1,5)(2,3)(4,13)(6,9)(7,10)(8,11)(12,15)(14,16)(17,22)(18,26)(19,21)(20,30)(23,31)(24,25)(27,29)(28,32)(33,39,40,43)(34,38,44,46)(35,47,48,42)(36,41,37,45)Polycyclic presentation:
2^9|a^2=gh,b^2=hi,c^2=gh,d^2=gi,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^d=egi,f^e=fgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10154841);