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 | |
4410 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(3,7)(5,8)(9,10,13,16)(11,14,18,21)(12,15,19,22)(17,20,23,24)(26,29)(27,33)(28,36)(39,40);(1,2)(3,5)(4,6)(7,8)(9,11,13,18)(10,14,16,21)(12,17,19,23)(15,20,22,24)(25,26,30,29)(27,31,33,35)(28,37,36,32)(34,40,38,39);(1,2,4,6)(3,5,7,8)(9,10)(11,14)(12,15)(13,16)(17,20)(18,21)(19,22)(23,24)(25,27,30,33)(26,31,29,35)(28,38,36,34)(32,40,37,39);(1,2)(3,5)(4,6)(7,8)(9,11)(10,14)(12,17)(13,18)(15,20)(16,21)(19,23)(22,24)(25,28,30,36)(26,32,29,37)(27,34,33,38)(31,39,35,40);(1,2)(3,5)(4,6)(7,8)(9,10)(11,14)(12,15)(13,16)(17,20)(18,21)(19,22)(23,24)(25,27)(26,31)(28,34)(29,35)(30,33)(32,39)(36,38)(37,40);(1,3)(2,5)(4,7)(6,8)(9,12,13,19)(10,15,16,22)(11,17,18,23)(14,20,21,24)(25,29,30,26)(27,35,33,31)(28,37,36,32)(34,40,38,39)Polycyclic presentation:
2^9|a^2=gh,b^2=h,c^2=gi,d^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,e^c=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8034311);