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 | |
4421 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,4)(6,8)(10,11)(12,19)(16,18)(20,21)(26,31)(32,33)(34,38)(39,40);(3,7)(6,8)(10,11)(15,17)(16,18)(23,24)(25,26)(27,32)(28,33)(29,34)(30,31)(35,39)(36,40)(37,38);(1,2,5,4)(3,6,7,8)(9,10,14,11)(12,15,19,17)(13,16,22,18)(20,23,21,24);(1,3)(2,6)(4,8)(5,7)(9,11,14,10)(12,17,19,15)(13,16,22,18)(20,23,21,24)(25,27,30,28)(26,32,31,33)(29,36,37,35)(34,40,38,39);(1,4,5,2)(3,8,7,6)(9,12,14,19)(10,15,11,17)(13,21,22,20)(16,24,18,23)(25,28,30,27)(26,33,31,32)(29,35,37,36)(34,39,38,40);(9,13,14,22)(10,16,11,18)(12,20,19,21)(15,23,17,24)(25,29,30,37)(26,34,31,38)(27,35,28,36)(32,39,33,40)Polycyclic presentation:
2^9|c^2=h,d^2=gi,e^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,f^d=fgi,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8595618);