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 | |
4287 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,6)(7,8)(9,10)(11,17)(12,16)(13,18)(14,15)(19,23)(20,22)(21,24)(25,26)(27,31)(28,36)(29,33)(30,35)(32,37)(34,38)(39,40);(1,2)(3,7)(4,8)(5,6)(12,20)(16,22)(19,24)(21,23)(25,27)(26,30)(28,34)(29,35)(31,33)(32,38)(36,39)(37,40);(10,15)(11,18)(16,22)(19,24)(25,27,29,35)(26,31,33,30)(28,34,37,40)(32,39,36,38);(1,3,5,4)(2,7,6,8)(9,11)(10,13)(12,19)(14,18)(15,17)(16,21)(20,24)(22,23)(25,27)(26,31)(28,34)(29,35)(30,33)(32,39)(36,38)(37,40);(9,12,14,20)(10,16,15,22)(11,19,18,24)(13,21,17,23)(25,26)(27,31)(28,32)(29,33)(30,35)(34,39)(36,37)(38,40);(1,4,5,3)(2,8,6,7)(9,13)(10,11)(12,21)(14,17)(15,18)(16,19)(20,23)(22,24)(25,28,29,37)(26,32,33,36)(27,34,35,40)(30,38,31,39)Polycyclic presentation:
2^9|c^2=i,d^2=gi,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dh,f^a=fhi,c^b=ci,e^b=ehi,d^c=dhi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7549312);