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 | |
4661 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)(3,6)(4,7)(5,8)(9,10,14,15)(11,17,12,16)(13,21,22,18)(19,23,20,24)(25,26,31,35)(27,34,38,32)(28,30,29,33)(36,37,39,40);(1,3,5,7)(2,6,8,4)(10,15)(16,17)(18,21)(23,24)(25,27)(26,32)(28,39)(29,36)(30,37)(31,38)(33,40)(34,35);(1,3)(2,4)(5,7)(6,8)(9,11,14,12)(10,16,15,17)(13,19,22,20)(18,23,21,24)(25,28,31,29)(26,33,35,30)(27,36,38,39)(32,37,34,40);(1,4,5,6)(2,7,8,3)(9,12,14,11)(10,17,15,16)(13,19,22,20)(18,23,21,24)(25,29,31,28)(26,30,35,33)(27,36,38,39)(32,37,34,40);(1,3)(2,4)(5,7)(6,8)(9,10)(11,16)(12,17)(13,18)(14,15)(19,23)(20,24)(21,22)(25,27)(26,34)(28,36)(29,39)(30,37)(31,38)(32,35)(33,40);(1,4)(2,3)(5,6)(7,8)(9,13)(10,18)(11,19)(12,20)(14,22)(15,21)(16,23)(17,24)(25,30)(26,28)(27,37)(29,35)(31,33)(32,39)(34,36)(38,40)Polycyclic presentation:
2^9|a^2=g,b^2=gh,c^2=g,d^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=egi,e^d=eghi,f^d=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10315698);