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 | |
4435 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,4)(3,7,8,6)(9,10,15,16)(11,13,12,17)(14,18,22,23)(19,21,20,24)(28,36)(31,37)(35,39)(38,40);(1,3,5,8)(2,6,4,7)(10,16)(11,12)(18,23)(19,20)(25,26)(27,34)(28,37)(29,32)(30,33)(31,36)(35,38)(39,40);(9,11,15,12)(10,17,16,13)(14,19,22,20)(18,24,23,21)(25,27,30,32)(26,29,33,34)(28,35,36,39)(31,38,37,40);(2,4)(6,7)(9,12,15,11)(10,13,16,17)(14,20,22,19)(18,21,23,24)(25,26)(27,29)(28,31)(30,33)(32,34)(35,38)(36,37)(39,40);(1,3)(2,6)(4,7)(5,8)(9,13)(10,11)(12,16)(14,21)(15,17)(18,19)(20,23)(22,24)(25,28)(26,31)(27,35)(29,38)(30,36)(32,39)(33,37)(34,40);(1,4,5,2)(3,7,8,6)(9,14,15,22)(10,18,16,23)(11,19,12,20)(13,21,17,24)(25,29)(26,32)(27,33)(28,38)(30,34)(31,39)(35,37)(36,40)Polycyclic presentation:
2^9|a^2=g,b^2=gh,c^2=i,d^2=h,f^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, 8386702);