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 |
| prime radical | #2464 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4092 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10)(11,16)(12,18)(13,19)(14,21)(15,17)(20,24)(22,23)(25,26,31,34)(27,35,37,29)(28,32,38,36)(30,33,39,40);(2,5)(6,7)(9,11,15,12)(10,16,17,18)(13,21,23,20)(14,22,24,19)(25,27,31,37)(26,29,34,35)(28,33,38,40)(30,36,39,32);(1,2)(3,6)(4,5)(7,8)(10,17)(11,12)(19,22)(20,21)(25,28,31,38)(26,32,34,36)(27,33,37,40)(29,30,35,39);(9,12,15,11)(10,18,17,16)(13,20,23,21)(14,19,24,22)(25,26)(27,29)(28,32)(30,33)(31,34)(35,37)(36,38)(39,40);(1,2)(3,7)(4,5)(6,8)(9,13)(10,19)(11,21)(12,20)(14,16)(15,23)(17,22)(18,24)(25,29)(26,27)(28,30)(31,35)(32,33)(34,37)(36,40)(38,39);(1,3)(2,6)(4,8)(5,7)(9,14,15,24)(10,20,17,21)(11,22,12,19)(13,18,23,16)(25,30)(26,33)(27,36)(28,35)(29,38)(31,39)(32,37)(34,40)Polycyclic presentation:
2^9|a^2=g,b^2=gh,c^2=g,d^2=h,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9688357);