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 | |
4095 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(2,6)(7,9)(8,12)(14,15)(17,18)(19,26)(20,28)(21,23)(22,24)(25,30)(27,31)(29,32)(33,34,40,39)(35,44,38,41)(36,43,37,42)(45,47,46,48);(1,2)(3,9)(4,8)(5,6)(7,10)(11,15)(12,13)(14,16)(17,19,23,22)(18,24,21,26)(20,29,30,27)(25,32,28,31)(33,35,40,38)(34,41,39,44)(36,46,37,45)(42,48,43,47);(3,10)(7,9)(11,16)(14,15)(18,21)(24,26)(25,28)(31,32)(33,36,40,37)(34,42,39,43)(35,45,38,46)(41,47,44,48);(1,2)(3,7)(4,12)(5,6)(8,13)(9,10)(11,15)(14,16)(17,20,23,30)(18,25,21,28)(19,27,22,29)(24,31,26,32)(33,37,40,36)(34,43,39,42)(35,46,38,45)(41,48,44,47);(1,3,5,10)(2,7,6,9)(4,11,13,16)(8,14,12,15)(17,21,23,18)(19,24,22,26)(20,28,30,25)(27,31,29,32)(33,38,40,35)(34,44,39,41)(36,46,37,45)(42,48,43,47);(1,4,5,13)(2,8,6,12)(3,11,10,16)(7,14,9,15)(17,22,23,19)(18,24,21,26)(20,27,30,29)(25,32,28,31)(33,39,40,34)(35,44,38,41)(36,43,37,42)(45,48,46,47)Polycyclic presentation:
2^9|a^2=i,b^2=h,c^2=i,d^2=h,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8509839);