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 | |
4035 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,5)(3,7,9,11)(4,8,12,13)(10,14,15,16)(17,18)(19,26)(20,24)(21,29)(22,23)(25,31)(27,28)(30,32)(33,34)(35,41)(36,46)(37,39)(38,42)(40,43)(44,48)(45,47);(1,3)(2,7)(4,10)(5,11)(6,9)(8,14)(12,15)(13,16)(18,23)(20,26)(25,29)(28,32)(33,35,40,37)(34,41,43,39)(36,45,38,44)(42,48,46,47);(3,9)(4,12)(7,11)(8,13)(17,19,22,24)(18,20,23,26)(21,30,31,27)(25,32,29,28)(33,36,40,38)(34,42,43,46)(35,44,37,45)(39,48,41,47);(1,4,6,12)(2,8,5,13)(3,10,9,15)(7,14,11,16)(17,19)(18,20)(21,27)(22,24)(23,26)(25,28)(29,32)(30,31)(33,37,40,35)(34,39,43,41)(36,44,38,45)(42,47,46,48);(1,5,6,2)(3,11,9,7)(4,8,12,13)(10,14,15,16)(17,20)(18,24)(19,23)(21,28)(22,26)(25,30)(27,29)(31,32)(33,38,40,36)(34,42,43,46)(35,45,37,44)(39,48,41,47);(1,3)(2,7)(4,10)(5,11)(6,9)(8,14)(12,15)(13,16)(17,21,22,31)(18,25,23,29)(19,27,24,30)(20,28,26,32)(33,39)(34,35)(36,47)(37,43)(38,48)(40,41)(42,45)(44,46)Polycyclic presentation:
2^9|a^2=ghi,b^2=gh,c^2=h,d^2=i,e^2=i,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,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, 8829742);