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 | |
4131 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,5,6)(3,7,9,10)(4,12,14,8)(11,16,13,15)(17,18,23,21)(19,24,25,20)(22,29,30,26)(27,28,32,31)(33,34)(35,46)(36,42)(37,47)(38,43)(39,44)(40,45)(41,48);(2,6)(3,9)(8,12)(11,13)(17,19)(18,24)(20,21)(22,27)(23,25)(26,31)(28,29)(30,32)(33,35,40,48)(34,41,45,46)(36,38,37,44)(39,47,43,42);(2,6)(7,10)(8,12)(15,16)(17,20,23,24)(18,25,21,19)(22,28,30,31)(26,32,29,27)(33,36,40,37)(34,42,45,47)(35,38,48,44)(39,46,43,41);(1,2)(3,10)(4,8)(5,6)(7,9)(11,16)(12,14)(13,15)(17,20)(18,19)(21,25)(22,28)(23,24)(26,27)(29,32)(30,31)(33,37,40,36)(34,42,45,47)(35,38,48,44)(39,41,43,46);(1,3,5,9)(2,7,6,10)(4,13,14,11)(8,16,12,15)(17,21,23,18)(19,24,25,20)(22,26,30,29)(27,28,32,31)(33,38)(34,43)(35,37)(36,48)(39,45)(40,44)(41,42)(46,47);(1,4,5,14)(2,8,6,12)(3,11,9,13)(7,15,10,16)(17,22)(18,26)(19,27)(20,28)(21,29)(23,30)(24,31)(25,32)(33,39,40,43)(34,44,45,38)(35,47,48,42)(36,46,37,41)Polycyclic presentation:
2^9|a^2=h,b^2=ghi,c^2=gh,d^2=ghi,e^2=h,f^2=g,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, 10154539);