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 | |
4523 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(3,8)(4,6)(9,10,14,15)(11,13,18,16)(12,17,20,23)(19,22,21,24)(25,26,30,31)(27,34,35,32)(28,33,38,29)(36,40,39,37);(1,2)(3,4)(5,7)(6,8)(10,15)(11,18)(17,23)(19,21)(26,31)(27,35)(28,38)(37,40);(1,3)(2,6)(4,7)(5,8)(10,15)(12,20)(13,16)(19,21)(28,38)(29,33)(36,39)(37,40);(1,2,5,7)(3,6,8,4)(9,11)(10,16)(12,21)(13,15)(14,18)(17,22)(19,20)(23,24)(25,27,30,35)(26,32,31,34)(28,39,38,36)(29,37,33,40);(1,2)(3,4)(5,7)(6,8)(9,12)(10,17)(11,19)(13,22)(14,20)(15,23)(16,24)(18,21)(25,28)(26,33)(27,36)(29,31)(30,38)(32,37)(34,40)(35,39);(1,4)(2,3)(5,6)(7,8)(9,13,14,16)(10,11,15,18)(12,22,20,24)(17,19,23,21)(25,29,30,33)(26,28,31,38)(27,37,35,40)(32,39,34,36)Polycyclic presentation:
2^9|a^2=g,d^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,e^c=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, 8947631);