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 | |
4211 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)(3,5)(4,8)(6,7)(9,10,15,19)(11,16,21,18)(12,22,20,17)(13,14,23,24)(25,26)(27,31)(28,34)(29,37)(30,33)(32,38)(35,40)(36,39);(2,7)(3,8)(9,11)(10,16)(12,23)(13,20)(14,17)(15,21)(18,19)(22,24)(25,27,30,28)(26,31,33,34)(29,35,38,36)(32,39,37,40);(1,3,6,8)(2,4,7,5)(9,12,15,20)(10,17,19,22)(11,13,21,23)(14,16,24,18)(25,28,30,27)(26,31,33,34)(29,36,38,35)(32,39,37,40);(1,4,6,5)(2,3,7,8)(9,11)(10,18)(12,13)(14,17)(15,21)(16,19)(20,23)(22,24);(1,2)(3,4)(5,8)(6,7)(9,13)(10,14)(11,20)(12,21)(15,23)(16,17)(18,22)(19,24)(25,26)(27,34)(28,31)(29,32)(30,33)(35,40)(36,39)(37,38);(1,5,6,4)(2,8,7,3)(9,14)(10,13)(11,17)(12,18)(15,24)(16,20)(19,23)(21,22)(25,29)(26,32)(27,35)(28,36)(30,38)(31,39)(33,37)(34,40)Polycyclic presentation:
2^9|a^2=gi,b^2=hi,c^2=h,d^2=g,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,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, 8833748);