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 | |
4442 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,8)(3,6,4,7)(11,18)(15,22)(19,23)(20,24)(25,26)(27,35)(28,32)(29,33)(30,34)(31,38)(36,39)(37,40);(1,2)(3,6)(4,7)(5,8)(9,10,14,17)(11,15,18,22)(12,13,16,21)(19,20,23,24)(25,27)(26,31)(28,36)(29,37)(30,38)(32,33)(34,35)(39,40);(9,11)(10,15)(12,19)(13,20)(14,18)(16,23)(17,22)(21,24)(25,26)(27,35)(28,32)(29,33)(30,34)(31,38)(36,39)(37,40);(1,3,5,4)(2,6,8,7)(9,11)(10,15)(12,19)(13,20)(14,18)(16,23)(17,22)(21,24)(25,28,30,37)(26,32,34,40)(27,36,38,29)(31,33,35,39);(1,4,5,3)(2,7,8,6)(9,12)(10,13)(11,19)(14,16)(15,20)(17,21)(18,23)(22,24)(25,26)(27,31)(28,32)(29,39)(30,34)(33,36)(35,38)(37,40);(9,13)(10,16)(11,20)(12,17)(14,21)(15,23)(18,24)(19,22)(25,29,30,36)(26,33,34,39)(27,37,38,28)(31,32,35,40)Polycyclic presentation:
2^9|a^2=gi,b^2=h,d^2=g,e^2=gi,f^2=i,b^a=bg,c^a=ch,d^a=dh,c^b=ci,f^e=fhi104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7561167);