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 | |
4660 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2)(3,7)(4,6)(5,8)(9,10)(11,14)(12,15)(13,16)(17,18)(19,21)(20,22)(23,24)(25,26,30,33)(27,31,36,32)(28,29,38,40)(34,35,37,39);(1,2,4,6)(3,5,8,7)(9,10,13,16)(11,14,17,18)(12,15,20,22)(19,21,23,24)(25,27)(26,31)(28,34)(29,35)(30,36)(32,33)(37,38)(39,40);(1,2)(3,5)(4,6)(7,8)(11,17)(12,20)(14,18)(15,22)(25,27,30,36)(26,32,33,31)(28,34,38,37)(29,39,40,35);(1,3)(2,5)(4,8)(6,7)(9,11)(10,14)(12,19)(13,17)(15,21)(16,18)(20,23)(22,24)(25,27)(26,32)(28,37)(29,35)(30,36)(31,33)(34,38)(39,40);(9,10)(11,18)(12,15)(13,16)(14,17)(19,24)(20,22)(21,23)(25,28)(26,29)(27,34)(30,38)(31,35)(32,39)(33,40)(36,37);(9,12)(10,15)(11,19)(13,20)(14,21)(16,22)(17,23)(18,24)(25,29)(26,28)(27,35)(30,40)(31,34)(32,37)(33,38)(36,39)Polycyclic presentation:
2^9|a^2=h,b^2=g,c^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dgi,f^c=fgi,e^d=egi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8834234);