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 | |
4705 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,6)(7,8)(10,14)(15,21)(16,19)(22,24)(25,26,31,33)(27,28,29,32)(30,34,38,40)(35,36,37,39);(1,2)(3,7)(4,8)(5,6)(9,10)(11,15)(12,19)(13,14)(16,20)(17,21)(18,24)(22,23)(25,27,31,29)(26,32,33,28)(30,37,38,35)(34,36,40,39);(10,14)(11,17)(16,19)(18,23)(26,33)(27,29)(34,40)(35,37);(9,10)(11,15)(12,16)(13,14)(17,21)(18,22)(19,20)(23,24)(25,28)(26,27)(29,33)(30,36)(31,32)(34,35)(37,40)(38,39);(1,3,5,4)(2,7,6,8)(9,11,13,17)(10,15,14,21)(12,18,20,23)(16,22,19,24)(25,29,31,27)(26,28,33,32)(30,37,38,35)(34,36,40,39);(1,4,5,3)(2,8,6,7)(9,12,13,20)(10,16,14,19)(11,18,17,23)(15,22,21,24)(25,30,31,38)(26,34,33,40)(27,35,29,37)(28,36,32,39)Polycyclic presentation:
2^9|a^2=h,b^2=h,e^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, 7646775);