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 | |
4401 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)(12,20)(16,21)(18,24)(22,23)(25,26,30,31)(27,32,35,28)(29,37,38,33)(34,36,40,39);(1,3,4,8)(2,5,7,6)(9,10)(11,17)(12,16)(13,14)(15,19)(18,23)(20,21)(22,24)(26,31)(28,32)(33,37)(36,39);(1,3)(2,6)(4,8)(5,7)(10,14)(15,17)(16,21)(22,23)(25,27,30,35)(26,32,31,28)(29,34,38,40)(33,39,37,36);(1,2)(3,6)(4,7)(5,8)(9,10)(11,15)(12,16)(13,14)(17,19)(18,22)(20,21)(23,24)(25,27)(26,28)(29,34)(30,35)(31,32)(33,36)(37,39)(38,40);(9,11,13,19)(10,15,14,17)(12,18,20,24)(16,22,21,23)(25,28)(26,27)(29,36)(30,32)(31,35)(33,34)(37,40)(38,39);(9,12)(10,16)(11,18)(13,20)(14,21)(15,22)(17,23)(19,24)(25,29,30,38)(26,33,31,37)(27,34,35,40)(28,36,32,39)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=g,e^2=hi,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fghi,c^b=ci,d^b=dh,e^b=eghi,d^c=dghi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8000724);