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 | #44930 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4508 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,6)(7,8)(9,10,15,14)(11,19,20,16)(12,17,21,22)(13,23,24,18);(1,2)(3,7)(4,8)(5,6)(9,11,15,20)(10,16,14,19)(12,13,21,24)(17,18,22,23)(25,26)(27,32)(28,33)(29,34)(30,31)(35,39)(36,37)(38,40);(1,3,5,4)(2,7,6,8)(10,14)(11,20)(12,21)(18,23)(26,31)(29,39)(32,37)(33,40);(1,3,5,4)(2,7,6,8)(9,12)(10,17)(11,13)(14,22)(15,21)(16,18)(19,23)(20,24)(25,27)(26,32)(28,34)(29,33)(30,36)(31,37)(35,38)(39,40);(1,4,5,3)(2,8,6,7)(9,13)(10,18)(11,21)(12,20)(14,23)(15,24)(16,22)(17,19)(25,28)(26,33)(27,35)(29,37)(30,38)(31,40)(32,39)(34,36);(9,14,15,10)(11,19,20,16)(12,22,21,17)(13,23,24,18)(25,29,30,39)(26,34,31,35)(27,33,36,40)(28,37,38,32)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=gh,d^2=gh,e^2=gh,f^2=i,b^a=bg,c^a=ch,e^a=eh,c^b=ci,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, 7686617);