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 | |
4509 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,4)(3,7,8,6)(9,10)(11,17)(12,20)(13,14)(15,18)(16,21)(19,22)(23,24)(25,26,31,28)(27,35,38,32)(29,33,37,39)(30,40,36,34);(1,3,5,8)(2,6,4,7)(9,10)(11,15)(12,16)(13,14)(17,18)(19,22)(20,21)(23,24)(25,27)(26,32)(28,35)(29,36)(30,37)(31,38)(33,40)(34,39);(1,2,5,4)(3,6,8,7)(10,14)(11,18)(16,20)(19,24)(25,26,31,28)(27,35,38,32)(29,39,37,33)(30,34,36,40);(1,2,5,4)(3,6,8,7)(9,10)(11,15)(12,16)(13,14)(17,18)(19,22)(20,21)(23,24)(25,28,31,26)(27,35,38,32)(29,39,37,33)(30,40,36,34);(1,4,5,2)(3,7,8,6)(9,11)(10,15)(12,19)(13,18)(14,17)(16,22)(20,23)(21,24)(25,29,31,37)(26,33,28,39)(27,36,38,30)(32,40,35,34);(9,12,13,21)(10,16,14,20)(11,19,18,24)(15,22,17,23)(25,30,31,36)(26,34,28,40)(27,37,38,29)(32,39,35,33)Polycyclic presentation:
2^9|a^2=g,b^2=ghi,c^2=g,d^2=g,e^2=g,f^2=i,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, 7619514);