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 | |
4503 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)(3,4)(5,6)(7,8)(9,10,14,17)(11,18,13,15)(12,20,22,16)(19,23,21,24)(25,26)(27,30)(28,29)(31,34)(32,38)(33,39)(35,40)(36,37);(2,6)(4,7)(9,11,14,13)(10,15,17,18)(12,19,22,21)(16,23,20,24)(25,27,31,33)(26,30,34,39)(28,36,38,35)(29,37,32,40);(9,12,14,22)(10,16,17,20)(11,19,13,21)(15,23,18,24)(25,28,31,38)(26,32,34,29)(27,35,33,36)(30,37,39,40);(1,3)(2,7)(4,6)(5,8)(9,12)(10,16)(11,19)(13,21)(14,22)(15,23)(17,20)(18,24)(25,28)(26,29)(27,36)(30,37)(31,38)(32,34)(33,35)(39,40);(1,4,5,7)(2,3,6,8)(9,13,14,11)(10,15,17,18)(12,21,22,19)(16,23,20,24)(25,29)(26,28)(27,37)(30,36)(31,32)(33,40)(34,38)(35,39);(1,3)(2,4)(5,8)(6,7)(9,10)(11,15)(12,16)(13,18)(14,17)(19,23)(20,22)(21,24)(25,30)(26,33)(27,34)(28,37)(29,35)(31,39)(32,36)(38,40)Polycyclic presentation:
2^9|a^2=hi,b^2=h,c^2=h,e^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9690202);