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 | |
4585 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,14,18)(11,17,22,15)(12,13,23,16)(19,24,20,21)(25,26,30,34)(27,31,28,32)(29,33,38,37)(35,39,36,40);(1,2)(3,8)(4,6)(5,7)(9,11)(10,15)(12,19)(13,21)(14,22)(16,24)(17,18)(20,23)(25,26)(27,31)(28,32)(29,33)(30,34)(35,39)(36,40)(37,38);(1,3,5,6)(2,4,7,8)(9,12)(10,16)(11,19)(13,18)(14,23)(15,24)(17,21)(20,22)(25,27,30,28)(26,31,34,32)(29,35,38,36)(33,39,37,40);(1,2)(3,4)(5,7)(6,8)(9,12)(10,16)(11,20)(13,18)(14,23)(15,21)(17,24)(19,22)(25,26)(27,31)(28,32)(29,37)(30,34)(33,38)(35,40)(36,39);(1,2)(3,4)(5,7)(6,8)(9,13,14,16)(10,12,18,23)(11,21,22,24)(15,19,17,20)(25,28,30,27)(26,32,34,31)(29,35,38,36)(33,39,37,40);(1,4)(2,6)(3,7)(5,8)(9,11)(10,17)(12,19)(13,24)(14,22)(15,18)(16,21)(20,23)(25,29,30,38)(26,33,34,37)(27,35,28,36)(31,39,32,40)Polycyclic presentation:
2^9|a^2=g,c^2=ghi,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^d=fgi,f^e=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8572506);