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 | |
4063 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,7)(3,5,10,8)(4,12,14,9)(11,15,16,13)(17,18)(19,24)(20,21)(22,30)(23,25)(26,31)(27,29)(28,32)(33,34)(35,40)(36,39)(37,43)(38,41)(42,47)(44,46)(45,48);(2,7)(5,8)(9,12)(13,15)(17,19,23,20)(18,24,25,21)(22,28,31,27)(26,29,30,32)(33,34,39,36)(35,40,37,43)(38,44,46,41)(42,48,45,47);(2,7)(3,10)(9,12)(11,16)(18,25)(19,20)(26,30)(27,28)(33,35,39,37)(34,40,36,43)(38,42,46,45)(41,47,44,48);(1,3)(2,8)(4,11)(5,7)(6,10)(9,15)(12,13)(14,16)(17,20,23,19)(18,21,25,24)(22,28,31,27)(26,29,30,32)(33,36,39,34)(35,40,37,43)(38,44,46,41)(42,47,45,48);(1,4)(2,9)(3,11)(5,13)(6,14)(7,12)(8,15)(10,16)(17,21)(18,19)(20,25)(22,29)(23,24)(26,27)(28,30)(31,32)(33,37,39,35)(34,40,36,43)(38,45,46,42)(41,47,44,48);(1,5,6,8)(2,3,7,10)(4,13,14,15)(9,11,12,16)(17,22,23,31)(18,26,25,30)(19,27,20,28)(21,29,24,32)(33,38)(34,41)(35,42)(36,44)(37,45)(39,46)(40,47)(43,48)Polycyclic presentation:
2^9|a^2=hi,b^2=gh,c^2=ghi,d^2=gh,e^2=ghi,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eghi,f^b=fgi,d^c=dgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8664657);