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 |
| prime radical | #2678 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4362 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)(11,20)(12,18)(13,14)(15,17)(16,21)(19,23)(22,24);(1,2,5,6)(3,7,4,8)(9,11)(10,16)(12,13)(14,24)(15,21)(17,20)(18,19)(22,23)(25,26)(27,31)(28,32)(29,30)(33,39)(34,35)(36,40)(37,38);(1,3,5,4)(2,7,6,8)(10,17)(11,21)(12,22)(14,19)(26,29)(31,34)(32,37)(39,40);(9,12,15,22)(10,18,17,24)(11,13,21,23)(14,16,19,20)(25,27,30,35)(26,31,29,34)(28,36,38,33)(32,40,37,39);(1,4,5,3)(2,8,6,7)(9,13)(10,19)(11,12)(14,17)(15,23)(16,18)(20,24)(21,22)(25,28,30,38)(26,32,29,37)(27,33,35,36)(31,39,34,40);(9,14,15,19)(10,13,17,23)(11,18,21,24)(12,16,22,20)(25,29,30,26)(27,34,35,31)(28,37,38,32)(33,40,36,39)Polycyclic presentation:
2^9|b^2=gh,c^2=gh,d^2=i,e^2=gi,f^2=i,b^a=bg,c^a=ch,e^a=eh,c^b=ci,f^b=fi,d^c=dh,f^c=fi,e^d=ei98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7756214);