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 | |
4778 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,3)(4,7,8,6)(9,10,14,15)(11,12,16,19)(13,17,22,23)(18,20,24,21)(25,26,31,33)(27,30,28,32)(29,34,38,40)(35,37,36,39);(2,3)(6,7)(10,15)(12,19)(17,23)(20,21)(25,27,31,28)(26,32,33,30)(29,36,38,35)(34,37,40,39);(1,3,5,2)(4,7,8,6)(26,33)(27,28)(34,40)(35,36);(1,4)(2,6)(3,7)(5,8)(9,11,14,16)(10,12,15,19)(13,18,22,24)(17,20,23,21)(25,28,31,27)(26,30,33,32)(29,36,38,35)(34,37,40,39);(9,12)(10,16)(11,15)(13,21)(14,19)(17,18)(20,22)(23,24)(25,29,31,38)(26,34,33,40)(27,35,28,36)(30,37,32,39);(9,13)(10,17)(11,18)(12,20)(14,22)(15,23)(16,24)(19,21)(25,30)(26,27)(28,33)(29,37)(31,32)(34,35)(36,40)(38,39)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=hi,d^2=ghi,e^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^e=fgi77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9830856);