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 | |
4555 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,5)(6,8)(12,19)(15,17)(20,24)(22,23)(25,26)(27,33)(28,35)(29,37)(30,31)(32,39)(34,36)(38,40);(3,7)(6,8)(9,10,14,11)(12,15,19,17)(13,18,21,16)(20,23,24,22)(25,27,30,35)(26,28,31,33)(29,38,39,34)(32,40,37,36);(1,2)(3,6)(4,5)(7,8)(10,11)(15,17)(16,18)(22,23)(26,31)(28,33)(32,37)(36,40);(9,11,14,10)(12,15,19,17)(13,18,21,16)(20,22,24,23)(25,28,30,33)(26,27,31,35)(29,36,39,40)(32,34,37,38);(1,3)(2,6)(4,7)(5,8)(9,12)(10,15)(11,17)(13,20)(14,19)(16,22)(18,23)(21,24)(25,27)(26,28)(29,34)(30,35)(31,33)(32,36)(37,40)(38,39);(1,2)(3,6)(4,5)(7,8)(9,13)(10,16)(11,18)(12,20)(14,21)(15,22)(17,23)(19,24)(25,29,30,39)(26,32,31,37)(27,34,35,38)(28,36,33,40)Polycyclic presentation:
2^9|b^2=gi,d^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^d=egi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8748663);