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 | |
4410 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,3)(4,7,8,6)(9,10,15,16)(11,17,12,20)(13,23,24,14)(18,22,21,19)(25,26,30,33)(27,28,35,31)(29,37,38,32)(34,39,40,36);(2,3)(6,7)(10,16)(13,24)(17,20)(18,21)(25,27,30,35)(26,31,33,28)(29,34,38,40)(32,39,37,36);(1,3,5,2)(4,7,8,6)(9,11,15,12)(10,17,16,20)(13,18,24,21)(14,19,23,22)(25,28,30,31)(26,27,33,35)(29,36,38,39)(32,34,37,40);(1,4,5,8)(2,6,3,7)(9,12,15,11)(10,17,16,20)(13,21,24,18)(14,19,23,22)(25,27)(26,31)(28,33)(29,34)(30,35)(32,39)(36,37)(38,40);(9,13)(10,14)(11,18)(12,21)(15,24)(16,23)(17,19)(20,22)(25,29,30,38)(26,32,33,37)(27,34,35,40)(28,36,31,39);(9,14)(10,13)(11,19)(12,22)(15,23)(16,24)(17,18)(20,21)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=g,d^2=gh,e^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,e^b=egi,f^d=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8380032);