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 | #38486 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4430 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)(11,16,14,12)(13,21,23,18)(19,20,22,24)(25,26,30,35)(27,36,38,31)(28,33,39,32)(29,40,37,34);(1,2,5,6)(3,7,4,8)(9,11,15,14)(10,16,17,12)(13,19,23,22)(18,24,21,20)(25,27,30,38)(26,31,35,36)(28,37,39,29)(32,34,33,40);(1,3,5,4)(2,7,6,8)(10,17)(12,16)(18,21)(20,24)(25,28,30,39)(26,32,35,33)(27,29,38,37)(31,40,36,34);(9,12)(10,11)(13,20)(14,17)(15,16)(18,19)(21,22)(23,24)(25,28)(26,33)(27,37)(29,38)(30,39)(31,40)(32,35)(34,36);(1,4,5,3)(2,8,6,7)(9,13,15,23)(10,18,17,21)(11,19,14,22)(12,20,16,24)(25,26)(27,31)(28,33)(29,34)(30,35)(32,39)(36,38)(37,40);(9,14,15,11)(10,12,17,16)(13,22,23,19)(18,20,21,24)(25,29,30,37)(26,34,35,40)(27,28,38,39)(31,33,36,32)Polycyclic presentation:
2^9|a^2=h,b^2=ghi,c^2=g,e^2=gh,f^2=h,b^a=bg,c^a=ch,e^a=eh,f^a=fi,c^b=ci,d^c=dh,e^c=ei104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7617004);