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 | |
4589 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,4)(6,8)(9,10)(11,16)(12,13)(14,15)(17,21)(18,24)(19,22)(20,23)(25,26)(27,32)(28,38)(29,33)(30,34)(31,35)(36,40)(37,39);(3,7)(6,8)(10,15)(12,21)(16,23)(18,22)(25,27)(26,32)(28,30)(29,39)(31,36)(33,37)(34,38)(35,40);(1,2,5,4)(3,6,7,8)(10,15)(13,17)(16,23)(19,24)(27,36)(30,37)(32,40)(34,39);(1,3,5,7)(2,6,4,8)(9,11,14,20)(10,16,15,23)(12,22,21,18)(13,19,17,24)(25,28)(26,29)(27,37)(30,36)(31,33)(32,34)(35,38)(39,40);(1,4,5,2)(3,8,7,6)(9,12)(10,17)(11,18)(13,15)(14,21)(16,19)(20,22)(23,24)(25,29)(26,33)(27,34)(28,35)(30,32)(31,38)(36,39)(37,40);(9,13,14,17)(10,12,15,21)(11,19,20,24)(16,22,23,18)(25,30,31,37)(26,34,35,39)(27,28,36,33)(29,32,38,40)Polycyclic presentation:
2^9|c^2=gh,d^2=h,e^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,f^c=fgi,e^d=egi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8325412);