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 | #41416 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4228 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,6)(7,8)(9,10,14,17)(11,15,18,22)(12,13,16,21)(19,20,23,24)(25,26,31,36)(27,32,39,33)(28,38,37,34)(29,35,40,30);(1,2)(3,7)(4,8)(5,6)(9,11)(10,15)(12,19)(13,20)(14,18)(16,23)(17,22)(21,24)(25,27,31,39)(26,32,36,33)(28,29,37,40)(30,38,35,34);(1,3,5,4)(2,7,6,8)(11,18)(15,22)(19,23)(20,24)(25,27,31,39)(26,33,36,32)(28,29,37,40)(30,34,35,38);(1,3,5,4)(2,7,6,8)(9,12)(10,13)(11,19)(14,16)(15,20)(17,21)(18,23)(22,24)(25,28)(26,34)(27,29)(30,32)(31,37)(33,35)(36,38)(39,40);(1,3,5,4)(2,7,6,8)(9,13)(10,16)(11,20)(12,17)(14,21)(15,23)(18,24)(19,22)(25,29)(26,35)(27,37)(28,39)(30,36)(31,40)(32,34)(33,38);(1,4,5,3)(2,8,6,7)(9,11)(10,15)(12,19)(13,20)(14,18)(16,23)(17,22)(21,24)(25,30,31,35)(26,29,36,40)(27,38,39,34)(28,32,37,33)Polycyclic presentation:
2^9|a^2=hi,b^2=h,c^2=gh,d^2=g,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dh,c^b=ci,f^c=fhi,e^d=ehi98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7557155);