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 | |
4466 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,15,18)(11,16,22,17)(12,23,13,14)(19,21,20,24)(26,31)(27,32)(29,37)(36,40);(1,2)(3,7)(4,5)(6,8)(9,11,15,22)(10,16,18,17)(12,20,13,19)(14,21,23,24)(25,26)(27,34)(28,32)(29,33)(30,31)(35,40)(36,39)(37,38);(2,5)(6,7)(9,12,15,13)(10,14,18,23)(11,19,22,20)(16,24,17,21)(25,27,30,32)(26,28,31,34)(29,35,37,39)(33,36,38,40);(2,5)(6,7)(9,13,15,12)(10,14,18,23)(11,20,22,19)(16,24,17,21)(25,26)(27,28)(29,33)(30,31)(32,34)(35,36)(37,38)(39,40);(1,2)(3,6)(4,5)(7,8)(9,14)(10,12)(11,21)(13,18)(15,23)(16,20)(17,19)(22,24)(25,28)(26,32)(27,31)(29,36)(30,34)(33,39)(35,38)(37,40);(1,3)(2,6)(4,8)(5,7)(9,11)(10,17)(12,19)(13,20)(14,21)(15,22)(16,18)(23,24)(25,29)(26,33)(27,35)(28,36)(30,37)(31,38)(32,39)(34,40)Polycyclic presentation:
2^9|a^2=gh,b^2=gh,c^2=h,d^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,f^b=fgi,e^c=egi,e^d=eghi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10228594);