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 | |
4555 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10)(11,13)(12,20)(14,16)(15,19)(17,23)(18,24)(21,22)(25,26,30,34)(27,28,32,31)(29,33,37,40)(35,36,39,38);(1,2,5,6)(3,7,8,4)(9,11)(10,15)(12,21)(13,16)(14,19)(17,22)(18,23)(20,24)(25,27,30,32)(26,31,34,28)(29,35,37,39)(33,38,40,36);(2,6)(4,7)(10,16)(13,15)(17,20)(22,24)(29,37)(33,40)(35,39)(36,38);(2,6)(3,8)(9,11)(10,15)(12,21)(13,16)(14,19)(17,22)(18,23)(20,24)(25,26)(27,28)(29,33)(30,34)(31,32)(35,36)(37,40)(38,39);(1,3)(2,4)(5,8)(6,7)(9,12,14,23)(10,17,16,20)(11,18,19,21)(13,22,15,24)(25,28)(26,32)(27,34)(29,36)(30,31)(33,39)(35,40)(37,38);(1,4)(2,3)(5,7)(6,8)(9,13,14,15)(10,11,16,19)(12,22,23,24)(17,18,20,21)(25,29,30,37)(26,33,34,40)(27,35,32,39)(28,36,31,38)Polycyclic presentation:
2^9|a^2=gh,b^2=ghi,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,f^c=fgi,e^d=egi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8295496);