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 | #48661 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4381 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10)(11,13)(12,15)(14,20)(16,18)(17,22)(19,23)(21,24)(25,26,31,35)(27,36,39,32)(28,37,38,33)(29,30,34,40);(1,2,4,6)(3,5,8,7)(9,11)(10,16)(12,13)(14,19)(15,18)(17,23)(20,21)(22,24)(25,27,31,39)(26,32,35,36)(28,30,38,40)(29,33,34,37);(1,2)(3,5)(4,6)(7,8)(10,12)(11,18)(17,20)(19,24);(9,12,15,10)(11,13,18,16)(14,20,22,17)(19,21,24,23)(25,28,31,38)(26,33,35,37)(27,30,39,40)(29,32,34,36);(1,2)(3,7)(4,6)(5,8)(9,13)(10,11)(12,18)(14,21)(15,16)(17,19)(20,24)(22,23)(25,29)(26,30)(27,37)(28,32)(31,34)(33,39)(35,40)(36,38);(1,3)(2,5)(4,8)(6,7)(9,14,15,22)(10,17,12,20)(11,19,18,24)(13,21,16,23)(25,30)(26,34)(27,38)(28,39)(29,35)(31,40)(32,37)(33,36)Polycyclic presentation:
2^9|a^2=gh,b^2=gi,d^2=g,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, 9695798);