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 | |
4519 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)(11,18)(12,13)(14,15)(16,21)(17,22)(19,20)(23,24)(25,26,30,28)(27,33,35,31)(29,32,37,36)(34,39,40,38);(1,2)(3,4)(5,6)(7,8)(10,15)(12,22)(16,18)(19,23)(25,27,30,35)(26,31,28,33)(29,34,37,40)(32,38,36,39);(2,6)(4,7)(10,15)(13,17)(16,18)(20,24)(29,37)(32,36)(34,40)(38,39);(2,6)(3,8)(9,11,14,21)(10,16,15,18)(12,23,22,19)(13,20,17,24)(25,27)(26,31)(28,33)(29,34)(30,35)(32,38)(36,39)(37,40);(1,3)(2,7)(4,6)(5,8)(9,12)(10,17)(11,19)(13,15)(14,22)(16,20)(18,24)(21,23)(25,28,30,26)(27,31,35,33)(29,36,37,32)(34,38,40,39);(1,4,5,7)(2,3,6,8)(9,13,14,17)(10,12,15,22)(11,20,21,24)(16,23,18,19)(25,29,30,37)(26,32,28,36)(27,34,35,40)(31,38,33,39)Polycyclic presentation:
2^9|a^2=gh,b^2=gh,d^2=h,e^2=gh,f^2=gi,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, 8306493);