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 | |
4345 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)(4,6)(5,8)(9,10)(11,18)(12,20)(13,15)(14,16)(17,21)(19,23)(22,24)(25,26,30,31)(27,34,29,32)(28,33,36,38)(35,40,37,39);(1,2,4,6)(3,5,8,7)(9,11,15,14)(10,16,13,18)(12,19,21,22)(17,23,20,24)(26,31)(28,36)(32,34)(35,37);(1,2)(3,5)(4,6)(7,8)(25,27,30,29)(26,32,31,34)(28,35,36,37)(33,39,38,40);(1,3)(2,5)(4,8)(6,7)(9,12)(10,17)(11,19)(13,20)(14,22)(15,21)(16,23)(18,24)(25,26)(27,32)(28,33)(29,34)(30,31)(35,39)(36,38)(37,40);(9,13,15,10)(11,16,14,18)(12,20,21,17)(19,23,22,24)(25,28)(26,33)(27,35)(29,37)(30,36)(31,38)(32,39)(34,40);(9,14,15,11)(10,16,13,18)(12,19,21,22)(17,24,20,23)(25,29,30,27)(26,32,31,34)(28,37,36,35)(33,39,38,40)Polycyclic presentation:
2^9|a^2=h,b^2=gh,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, 8413217);