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 | |
4702 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,5)(6,8)(9,10,14,15)(11,12,19,16)(13,17,22,23)(18,20,21,24)(25,26)(27,31)(28,29)(30,33)(32,36)(34,39)(35,37)(38,40);(3,7)(6,8)(10,15)(12,16)(17,23)(20,24)(25,27,30,36)(26,31,33,32)(28,37,38,34)(29,35,40,39);(1,2)(3,6)(4,5)(7,8)(25,27)(26,32)(28,34)(29,35)(30,36)(31,33)(37,38)(39,40);(1,3)(2,6)(4,7)(5,8)(9,11)(10,16)(12,15)(13,21)(14,19)(17,20)(18,22)(23,24)(25,27)(26,31)(28,34)(29,39)(30,36)(32,33)(35,40)(37,38);(1,2)(3,6)(4,5)(7,8)(9,12,14,16)(10,11,15,19)(13,20,22,24)(17,18,23,21)(25,28)(26,29)(27,34)(30,38)(31,39)(32,35)(33,40)(36,37);(9,13)(10,17)(11,18)(12,20)(14,22)(15,23)(16,24)(19,21)(25,29)(26,28)(27,35)(30,40)(31,37)(32,34)(33,38)(36,39)Polycyclic presentation:
2^9|a^2=g,b^2=i,e^2=g,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, 8382394);