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 | |
4408 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)(3,7)(4,5)(6,8)(9,10)(11,17)(12,14)(13,16)(15,19)(18,24)(20,22)(21,23)(25,26,30,31)(27,34,29,32)(28,33,36,38)(35,40,37,39);(2,5)(6,7)(10,12)(15,17)(16,20)(23,24)(26,31)(28,36)(32,34)(35,37);(1,2)(3,6)(4,5)(7,8)(25,27,30,29)(26,32,31,34)(28,35,36,37)(33,39,38,40);(1,3)(2,6)(4,8)(5,7)(9,11)(10,15)(12,17)(13,21)(14,19)(16,24)(18,22)(20,23)(25,26)(27,32)(28,33)(29,34)(30,31)(35,39)(36,38)(37,40);(9,12,14,10)(11,17,19,15)(13,20,22,16)(18,24,21,23)(25,28)(26,33)(27,35)(29,37)(30,36)(31,38)(32,39)(34,40);(9,13,14,22)(10,16,12,20)(11,18,19,21)(15,23,17,24)(25,29,30,27)(26,32,31,34)(28,37,36,35)(33,39,38,40)Polycyclic presentation:
2^9|a^2=h,c^2=h,e^2=ghi,f^2=gi,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, 8413187);