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 | |
4906 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,7)(3,4,8,6)(10,14)(15,18)(16,20)(22,24)(25,26)(27,32)(28,34)(29,35)(30,37)(31,33)(36,40)(38,39);(1,3,5,8)(2,6,7,4)(9,10,13,14)(11,15,17,18)(12,20,21,16)(19,24,23,22)(25,27,31,29)(26,32,33,35)(28,37,38,36)(30,39,40,34);(1,3)(2,4)(5,8)(6,7)(11,17)(15,18)(19,23)(22,24)(26,33)(27,29)(34,39)(36,37);(1,4)(2,3)(5,6)(7,8)(9,10)(11,18)(12,16)(13,14)(15,17)(19,24)(20,21)(22,23)(25,28)(26,34)(27,36)(29,37)(30,35)(31,38)(32,40)(33,39);(9,11)(10,15)(12,19)(13,17)(14,18)(16,22)(20,24)(21,23)(25,29,31,27)(26,35,33,32)(28,37,38,36)(30,39,40,34);(9,12)(10,16)(11,19)(13,21)(14,20)(15,22)(17,23)(18,24)(25,30)(26,36)(27,34)(28,35)(29,39)(31,40)(32,38)(33,37)Polycyclic presentation:
2^9|a^2=hi,b^2=gh,e^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,f^b=fgi,e^c=egi,e^d=eghi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10232789);