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 | |
4396 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(3,7)(6,8)(9,10,14,15)(11,18,16,12)(13,21,22,17)(19,20,23,24)(25,26,30,34)(27,33,29,31)(28,32,36,39)(35,40,37,38);(1,2)(3,6)(4,5)(7,8)(10,15)(11,16)(17,21)(19,23)(25,27,30,29)(26,31,34,33)(28,35,36,37)(32,38,39,40);(2,5)(6,8)(10,15)(12,18)(17,21)(20,24)(28,36)(32,39)(35,37)(38,40);(9,11,14,16)(10,12,15,18)(13,19,22,23)(17,20,21,24)(25,26)(27,31)(28,32)(29,33)(30,34)(35,38)(36,39)(37,40);(1,2)(3,6)(4,5)(7,8)(9,12)(10,16)(11,15)(13,20)(14,18)(17,23)(19,21)(22,24)(25,28)(26,32)(27,35)(29,37)(30,36)(31,38)(33,40)(34,39);(1,3)(2,6)(4,7)(5,8)(9,13,14,22)(10,17,15,21)(11,19,16,23)(12,20,18,24)(25,29,30,27)(26,33,34,31)(28,37,36,35)(32,40,39,38)Polycyclic presentation:
2^9|a^2=g,b^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, 8023062);