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 | |
4466 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,6)(7,8)(9,10)(11,16)(12,17)(13,18)(14,15)(19,24)(20,21)(22,23)(26,32)(29,35)(33,36)(37,39);(1,2)(3,7)(4,8)(5,6)(25,26)(27,36)(28,33)(29,34)(30,35)(31,32)(37,40)(38,39);(10,15)(13,16)(17,23)(21,24)(25,27,31,28)(26,33,32,36)(29,39,35,37)(30,38,34,40);(1,3,5,4)(2,7,6,8)(9,11,14,18)(10,16,15,13)(12,20,22,19)(17,21,23,24)(25,28,31,27)(26,33,32,36)(29,37,35,39)(30,38,34,40);(1,3,5,4)(2,7,6,8)(9,12)(10,17)(11,19)(13,21)(14,22)(15,23)(16,24)(18,20)(25,29,31,35)(26,34,32,30)(27,37,28,39)(33,38,36,40);(1,4,5,3)(2,8,6,7)(9,13,14,16)(10,18,15,11)(12,21,22,24)(17,20,23,19)(25,30,31,34)(26,35,32,29)(27,38,28,40)(33,37,36,39)Polycyclic presentation:
2^9|c^2=i,d^2=gh,e^2=g,f^2=gh,b^a=bg,c^a=ch,e^a=ei,c^b=ci,e^c=ei,f^c=fh,e^d=ehi,f^d=fi95 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7982706);