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 | |
4558 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,6)(3,8)(10,15)(11,17)(12,19)(20,24)(25,26,31,36)(27,32,38,40)(28,30,35,33)(29,34,37,39);(1,2,5,6)(3,4,8,7)(9,10,14,15)(11,13,17,21)(12,16,19,22)(18,20,23,24)(25,27,31,38)(26,32,36,40)(28,29,35,37)(30,34,33,39);(1,3)(2,4)(5,8)(6,7)(25,28,31,35)(26,33,36,30)(27,37,38,29)(32,34,40,39);(1,3)(2,7)(4,6)(5,8)(9,11)(10,13)(12,18)(14,17)(15,21)(16,20)(19,23)(22,24)(25,27)(26,32)(28,37)(29,35)(30,39)(31,38)(33,34)(36,40);(9,12)(10,16)(11,18)(13,20)(14,19)(15,22)(17,23)(21,24)(25,29,31,37)(26,34,36,39)(27,35,38,28)(30,32,33,40);(1,4,5,7)(2,8,6,3)(9,13)(10,17)(11,15)(12,20)(14,21)(16,23)(18,22)(19,24)(25,30,31,33)(26,35,36,28)(27,34,38,39)(29,32,37,40)Polycyclic presentation:
2^9|a^2=i,b^2=gi,c^2=i,e^2=i,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,f^d=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8364231);