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 | |
4187 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)(3,10)(4,8)(5,13)(6,7)(9,14)(11,12)(15,16)(17,18,23,25)(19,24,29,31)(20,26,30,32)(21,27,28,22)(34,39)(35,41)(40,47)(42,43);(2,7)(4,10)(9,13)(12,16)(17,19)(18,24)(20,28)(21,30)(22,26)(23,29)(25,31)(27,32)(33,34,38,39)(35,37,41,46)(36,40,45,47)(42,44,43,48);(2,7)(3,8)(5,14)(12,16)(18,25)(19,29)(20,30)(22,27)(35,41)(37,46)(42,43)(44,48);(1,3)(2,4)(5,11)(6,8)(7,10)(9,12)(13,16)(14,15)(17,20,23,30)(18,26,25,32)(19,28,29,21)(22,31,27,24)(33,35,38,41)(34,37,39,46)(36,43,45,42)(40,48,47,44);(1,4)(2,8)(3,7)(5,12)(6,10)(9,15)(11,13)(14,16)(17,21,23,28)(18,27,25,22)(19,20,29,30)(24,26,31,32)(33,36)(34,40)(35,42)(37,44)(38,45)(39,47)(41,43)(46,48);(1,5,6,14)(2,9,7,13)(3,11,8,15)(4,12,10,16)(17,22,23,27)(18,28,25,21)(19,26,29,32)(20,31,30,24)(33,37)(34,41)(35,39)(36,44)(38,46)(40,43)(42,47)(45,48)Polycyclic presentation:
2^9|a^2=i,b^2=ghi,d^2=gh,e^2=i,f^2=h,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, 8827929);