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 | |
4477 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,3)(4,7,8,6)(9,10)(11,20)(12,18)(13,19)(14,21)(15,17)(16,22)(23,24)(27,33)(31,38)(34,36)(39,40);(2,3)(6,7)(9,11,15,22)(10,16,17,20)(12,13,21,23)(14,19,18,24)(25,26)(27,31)(28,35)(29,30)(32,37)(33,38)(34,40)(36,39);(1,3,5,2)(4,7,8,6)(10,17)(14,18)(16,20)(19,24)(26,30)(31,38)(32,35)(39,40);(1,4)(2,6)(3,7)(5,8)(9,12)(10,18)(11,13)(14,17)(15,21)(16,24)(19,20)(22,23)(25,27)(26,31)(28,36)(29,33)(30,38)(32,40)(34,37)(35,39);(9,13)(10,19)(11,21)(12,22)(14,20)(15,23)(16,18)(17,24)(25,28)(26,32)(27,34)(29,37)(30,35)(31,39)(33,36)(38,40);(9,14,15,18)(10,12,17,21)(11,19,22,24)(13,20,23,16)(25,26)(27,31)(28,32)(29,30)(33,38)(34,39)(35,37)(36,40)Polycyclic presentation:
2^9|a^2=gh,b^2=h,c^2=gh,f^2=h,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, 8833227);