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 | |
4485 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,5)(6,8)(9,10,15,17)(11,19,13,16)(12,18,22,14)(20,21,23,24)(25,26,31,28)(27,32,35,36)(29,39,38,33)(30,40,37,34);(1,2)(3,6)(4,5)(7,8)(9,11,15,13)(10,16,17,19)(12,20,22,23)(14,21,18,24)(25,27)(26,32)(28,36)(29,37)(30,38)(31,35)(33,40)(34,39);(10,17)(12,22)(16,19)(20,23)(26,28)(27,35)(29,38)(34,40);(1,3,4,7)(2,6,5,8)(9,12)(10,18)(11,20)(13,23)(14,17)(15,22)(16,24)(19,21)(25,28,31,26)(27,36,35,32)(29,39,38,33)(30,40,37,34);(9,13,15,11)(10,19,17,16)(12,23,22,20)(14,24,18,21)(25,29,31,38)(26,33,28,39)(27,37,35,30)(32,40,36,34);(9,14,15,18)(10,12,17,22)(11,21,13,24)(16,20,19,23)(25,30,31,37)(26,34,28,40)(27,38,35,29)(32,39,36,33)Polycyclic presentation:
2^9|a^2=h,b^2=hi,d^2=gh,e^2=h,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, 7617093);