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 | |
4486 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,7)(3,4,8,6)(9,10)(11,19)(12,17)(13,18)(14,15)(16,20)(21,22)(23,24)(25,26)(27,33)(28,32)(29,30)(31,35)(34,40)(36,38)(37,39);(1,3,5,8)(2,6,7,4)(10,15)(13,23)(16,19)(17,21)(26,29)(28,36)(31,33)(34,37);(1,3)(2,4)(5,8)(6,7)(11,20)(12,22)(16,19)(17,21)(25,27,30,35)(26,31,29,33)(28,37,36,34)(32,40,38,39);(1,4)(2,3)(5,6)(7,8)(9,11)(10,16)(12,18)(13,21)(14,20)(15,19)(17,23)(22,24)(25,27)(26,31)(28,34)(29,33)(30,35)(32,39)(36,37)(38,40);(9,12)(10,17)(11,18)(13,19)(14,22)(15,21)(16,23)(20,24)(25,28)(26,32)(27,34)(29,38)(30,36)(31,39)(33,40)(35,37);(9,13,14,23)(10,18,15,24)(11,21,20,17)(12,19,22,16)(25,29,30,26)(27,33,35,31)(28,38,36,32)(34,40,37,39)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,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, 8717496);