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 | |
4512 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10)(11,19)(12,21)(13,18)(14,15)(16,22)(17,23)(20,24)(25,26,30,33)(27,28,35,31)(29,32,37,39)(34,36,40,38);(1,2,5,6)(3,7,8,4)(10,15)(12,23)(16,19)(18,20)(25,27,30,35)(26,31,33,28)(29,34,37,40)(32,38,39,36);(2,6)(4,7)(10,15)(13,24)(16,19)(17,21)(29,37)(32,39)(34,40)(36,38);(2,6)(3,8)(9,11,14,22)(10,16,15,19)(12,18,23,20)(13,21,24,17)(25,28,30,31)(26,27,33,35)(29,36,37,38)(32,34,39,40);(1,3)(2,4)(5,8)(6,7)(9,12,14,23)(10,17,15,21)(11,20,22,18)(13,16,24,19)(25,27)(26,28)(29,34)(30,35)(31,33)(32,36)(37,40)(38,39);(1,4)(2,3)(5,7)(6,8)(9,13,14,24)(10,18,15,20)(11,21,22,17)(12,16,23,19)(25,29,30,37)(26,32,33,39)(27,34,35,40)(28,36,31,38)Polycyclic presentation:
2^9|a^2=gh,b^2=ghi,d^2=g,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,f^c=fgi,e^d=egi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8295504);