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 | #2541 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4232 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,14,15)(11,16,18,12)(13,21,23,17)(19,20,24,22)(25,26,30,34)(27,29,33,31)(28,32,38,40)(35,39,36,37);(1,2,5,7)(3,8,4,6)(9,10,14,15)(11,16,18,12)(13,17,23,21)(19,22,24,20)(25,27,30,33)(26,31,34,29)(28,36,38,35)(32,39,40,37);(1,3,5,4)(2,6,7,8)(10,15)(11,18)(17,21)(19,24);(9,11)(10,16)(12,15)(13,19)(14,18)(17,22)(20,21)(23,24)(25,26)(27,31)(28,32)(29,33)(30,34)(35,37)(36,39)(38,40);(1,2)(3,6)(4,8)(5,7)(9,12,14,16)(10,11,15,18)(13,22,23,20)(17,24,21,19)(25,28,30,38)(26,32,34,40)(27,35,33,36)(29,39,31,37);(1,4,5,3)(2,6,7,8)(9,13)(10,17)(11,19)(12,20)(14,23)(15,21)(16,22)(18,24)(25,29,30,31)(26,33,34,27)(28,37,38,39)(32,35,40,36)Polycyclic presentation:
2^9|a^2=gh,b^2=gi,c^2=hi,e^2=gh,f^2=ghi,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, 9796644);