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 | |
4141 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,10)(3,8,4,7)(5,9,13,16)(11,15,12,14)(17,18)(19,26)(20,23)(21,28)(22,24)(25,30)(27,31)(29,32)(33,34)(35,44)(36,39)(37,47)(38,42)(40,43)(41,46)(45,48);(1,3,6,4)(2,7,10,8)(5,11,13,12)(9,14,16,15)(17,19,23,22)(18,24,20,26)(21,27,30,29)(25,31,28,32)(33,35,40,46)(34,41,43,44)(36,42,37,48)(38,39,45,47);(1,3,6,4)(2,7,10,8)(5,12,13,11)(9,15,16,14)(17,18,23,20)(19,26,22,24)(21,25,30,28)(27,32,29,31)(33,36,40,37)(34,39,43,47)(35,42,46,48)(38,41,45,44);(1,4,6,3)(2,8,10,7)(5,12,13,11)(9,15,16,14)(17,20,23,18)(19,24,22,26)(21,28,30,25)(27,31,29,32)(33,37,40,36)(34,39,43,47)(35,42,46,48)(38,44,45,41);(1,2)(3,7)(4,8)(5,9)(6,10)(11,14)(12,15)(13,16)(17,21)(18,25)(19,27)(20,28)(22,29)(23,30)(24,31)(26,32)(33,38,40,45)(34,42,43,48)(35,39,46,47)(36,41,37,44);(1,5,6,13)(2,9,10,16)(3,11,4,12)(7,14,8,15)(17,22,23,19)(18,24,20,26)(21,29,30,27)(25,31,28,32)(33,39)(34,36)(35,45)(37,43)(38,46)(40,47)(41,42)(44,48)Polycyclic presentation:
2^9|a^2=hi,b^2=g,c^2=g,d^2=g,e^2=gh,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^c=fgi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8087582);