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 | |
4353 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)(3,5)(4,7)(6,8)(9,10)(11,14)(12,16)(13,17)(15,18)(19,22)(20,21)(23,24)(27,35)(28,37)(31,38)(32,40);(2,3)(6,7)(9,11)(10,14)(12,19)(13,18)(15,17)(16,22)(20,23)(21,24)(25,26,30,34)(27,31,35,38)(28,32,37,40)(29,33,36,39);(1,2,5,3)(4,6,8,7)(11,18)(14,15)(19,24)(22,23)(25,27,30,35)(26,31,34,38)(28,36,37,29)(32,39,40,33);(1,3,5,2)(4,7,8,6)(9,11)(10,15)(12,19)(13,18)(14,17)(16,23)(20,22)(21,24)(25,26)(27,31)(28,32)(29,33)(30,34)(35,38)(36,39)(37,40);(1,4,5,8)(2,6,3,7)(9,12)(10,16)(11,19)(13,21)(14,22)(15,23)(17,20)(18,24)(25,28)(26,32)(27,36)(29,35)(30,37)(31,39)(33,38)(34,40);(9,10)(11,14)(12,20)(13,17)(15,18)(16,21)(19,23)(22,24)(25,29,30,36)(26,33,34,39)(27,28,35,37)(31,32,38,40)Polycyclic presentation:
2^9|b^2=gh,c^2=h,d^2=g,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8636254);