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 | |
4329 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)(6,8)(10,14)(12,19)(15,20)(17,23)(25,26)(27,28)(29,32)(30,37)(31,33)(34,39)(35,40)(36,38);(1,2)(3,6)(4,5)(7,8)(9,10)(11,15)(12,16)(13,14)(17,22)(18,20)(19,21)(23,24)(25,27,32,31)(26,33,29,28)(30,35,39,38)(34,40,37,36);(2,5)(6,8)(9,11,13,18)(10,15,14,20)(12,17,19,23)(16,22,21,24)(25,28)(26,31)(27,29)(30,36)(32,33)(34,38)(35,37)(39,40);(9,10)(11,15)(12,16)(13,14)(17,22)(18,20)(19,21)(23,24)(25,29,32,26)(27,33,31,28)(30,37,39,34)(35,40,38,36);(1,2)(3,6)(4,5)(7,8)(9,11)(10,15)(12,17)(13,18)(14,20)(16,22)(19,23)(21,24)(25,30)(26,34)(27,35)(28,36)(29,37)(31,38)(32,39)(33,40);(1,3)(2,6)(4,7)(5,8)(9,12,13,19)(10,16,14,21)(11,17,18,23)(15,22,20,24)(25,31,32,27)(26,33,29,28)(30,38,39,35)(34,40,37,36)Polycyclic presentation:
2^9|b^2=h,c^2=gh,d^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^c=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8023133);