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 | |
4435 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)(11,14)(12,15)(13,16)(17,19)(18,21)(20,22)(23,24)(25,26,30,28)(27,33,35,31)(29,36,38,32)(34,39,37,40);(1,2,5,3)(4,6,8,7)(10,11)(15,17)(16,18)(22,23)(25,27,30,35)(26,31,28,33)(29,34,38,37)(32,39,36,40);(2,3)(6,7)(12,19)(15,17)(20,24)(22,23)(26,28)(31,33)(32,36)(39,40);(2,3)(4,8)(9,11,14,10)(12,15,19,17)(13,16,21,18)(20,23,24,22)(25,27)(26,31)(28,33)(29,37)(30,35)(32,40)(34,38)(36,39);(1,3,5,2)(4,7,8,6)(9,12)(10,15)(11,17)(13,20)(14,19)(16,22)(18,23)(21,24)(25,28,30,26)(27,31,35,33)(29,36,38,32)(34,39,37,40);(1,4)(2,6)(3,7)(5,8)(9,13,14,21)(10,16,11,18)(12,20,19,24)(15,22,17,23)(25,29,30,38)(26,32,28,36)(27,34,35,37)(31,39,33,40)Polycyclic presentation:
2^9|a^2=h,b^2=hi,d^2=gh,e^2=hi,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,e^c=egi,e^d=egi,f^d=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9012461);