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 | |
4592 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,6,7)(3,5,4,8)(9,10,14,18)(11,13,17,15)(12,16,21,24)(19,20,23,22)(26,32)(27,35)(28,37)(34,40);(2,7)(3,4)(9,11,14,17)(10,13,18,15)(12,19,21,23)(16,20,24,22)(25,26,31,32)(27,33,35,39)(28,38,37,30)(29,34,36,40);(1,3,6,4)(2,8,7,5)(9,11)(10,15)(12,19)(13,18)(14,17)(16,22)(20,24)(21,23)(25,27)(26,33)(28,36)(29,37)(30,34)(31,35)(32,39)(38,40);(1,4,6,3)(2,5,7,8)(9,11)(10,13)(12,19)(14,17)(15,18)(16,20)(21,23)(22,24)(25,28,31,37)(26,30,32,38)(27,36,35,29)(33,34,39,40);(1,5)(2,3)(4,7)(6,8)(9,12,14,21)(10,16,18,24)(11,19,17,23)(13,20,15,22)(25,29,31,36)(26,34,32,40)(27,37,35,28)(30,39,38,33);(9,13)(10,17)(11,18)(12,20)(14,15)(16,23)(19,24)(21,22)(25,30)(26,28)(27,34)(29,39)(31,38)(32,37)(33,36)(35,40)Polycyclic presentation:
2^9|a^2=i,b^2=gi,c^2=ghi,d^2=g,e^2=gi,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, 8363107);