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 | #43117 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4414 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)(9,10,15,17)(11,19,13,16)(12,18,22,14)(20,21,23,24)(25,26)(27,37)(28,33)(29,34)(30,35)(31,36)(32,38)(39,40);(1,2)(3,7)(4,8)(5,6)(9,11,15,13)(10,16,17,19)(12,20,22,23)(14,21,18,24)(25,27)(26,32)(28,39)(29,40)(30,34)(31,38)(33,35)(36,37);(10,17)(12,22)(16,19)(20,23)(27,38)(30,39)(32,37)(35,40);(9,12)(10,18)(11,20)(13,23)(14,17)(15,22)(16,24)(19,21)(25,28)(26,33)(27,39)(29,36)(30,38)(31,34)(32,35)(37,40);(1,3,5,4)(2,7,6,8)(9,13,15,11)(10,19,17,16)(12,23,22,20)(14,24,18,21)(25,29,31,33)(26,34,36,28)(27,40,38,35)(30,37,39,32);(1,4,5,3)(2,8,6,7)(9,14,15,18)(10,12,17,22)(11,21,13,24)(16,20,19,23)(25,30,31,39)(26,35,36,40)(27,28,38,34)(29,37,33,32)Polycyclic presentation:
2^9|a^2=h,b^2=h,e^2=g,f^2=g,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, 7719289);