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 | |
4755 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2)(3,7)(4,8)(5,6)(9,10,13,16)(11,14,17,18)(12,15,20,22)(19,21,23,24)(25,26,31,34)(27,29,28,32)(30,38,39,33)(35,40,36,37);(2,6)(3,8)(9,10)(11,14)(12,15)(13,16)(17,18)(19,21)(20,22)(23,24)(25,27,31,28)(26,32,34,29)(30,35,39,36)(33,40,38,37);(1,3)(2,4)(5,8)(6,7)(11,17)(12,20)(14,18)(15,22)(25,27,31,28)(26,32,34,29)(30,36,39,35)(33,37,38,40);(1,4)(2,3)(5,7)(6,8)(9,10)(11,18)(12,15)(13,16)(14,17)(19,24)(20,22)(21,23)(25,28,31,27)(26,29,34,32)(30,36,39,35)(33,37,38,40);(1,2)(3,4)(5,6)(7,8)(9,11)(10,14)(12,19)(13,17)(15,21)(16,18)(20,23)(22,24)(25,29)(26,28)(27,34)(30,37)(31,32)(33,36)(35,38)(39,40);(1,4)(2,3)(5,7)(6,8)(9,12)(10,15)(11,19)(13,20)(14,21)(16,22)(17,23)(18,24)(25,30,31,39)(26,33,34,38)(27,35,28,36)(29,37,32,40)Polycyclic presentation:
2^9|a^2=gi,b^2=hi,c^2=hi,d^2=hi,f^2=hi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,e^c=egi,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, 8875495);