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 | |
4544 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)(4,5)(6,8)(9,10)(11,15)(12,16)(13,14)(17,19)(18,22)(20,21)(23,24)(25,26,30,32)(27,34,38,31)(28,33,35,29)(36,37,39,40);(2,5)(6,7)(10,14)(15,17)(16,21)(22,23)(25,27)(26,31)(28,36)(29,37)(30,38)(32,34)(33,40)(35,39);(1,2)(3,6)(4,5)(7,8)(12,20)(16,21)(18,24)(22,23)(26,32)(29,33)(31,34)(37,40);(1,3,4,8)(2,6,5,7)(9,11,13,19)(10,15,14,17)(12,18,20,24)(16,22,21,23)(25,28,30,35)(26,33,32,29)(27,36,38,39)(31,40,34,37);(9,10)(11,17)(12,16)(13,14)(15,19)(18,23)(20,21)(22,24)(25,27)(26,34)(28,39)(29,37)(30,38)(31,32)(33,40)(35,36);(9,12)(10,16)(11,18)(13,20)(14,21)(15,22)(17,23)(19,24)(25,29,30,33)(26,35,32,28)(27,37,38,40)(31,39,34,36)Polycyclic presentation:
2^9|a^2=h,d^2=g,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,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, 8833132);