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 | |
4719 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,8)(3,6,4,7)(9,10)(11,14)(12,16)(13,17)(15,18)(19,21)(20,23)(22,24)(25,26)(27,36)(28,39)(29,30)(31,33)(32,37)(34,38)(35,40);(1,2)(3,6)(4,7)(5,8)(9,11)(10,14)(12,19)(13,18)(15,17)(16,21)(20,24)(22,23)(25,27,31,37)(26,32,33,36)(28,29,38,40)(30,39,35,34);(11,18)(12,20)(14,15)(16,23)(26,33)(27,37)(29,40)(34,39);(1,3,5,4)(2,6,8,7)(9,11)(10,15)(12,19)(13,18)(14,17)(16,22)(20,24)(21,23)(25,28,31,38)(26,34,33,39)(27,29,37,40)(30,36,35,32);(9,12)(10,16)(11,19)(13,20)(14,21)(15,22)(17,23)(18,24)(25,29)(26,35)(27,38)(28,37)(30,33)(31,40)(32,34)(36,39);(1,4,5,3)(2,7,8,6)(9,10)(11,15)(12,16)(13,17)(14,18)(19,22)(20,23)(21,24)(25,30,31,35)(26,29,33,40)(27,34,37,39)(28,36,38,32)Polycyclic presentation:
2^9|a^2=gh,b^2=h,d^2=g,f^2=g,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,f^b=fi,d^c=dhi,e^c=ei92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7646772);