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 |
| prime radical | #37198 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4456 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(4,8)(6,7)(9,10)(11,18)(12,13)(14,15)(16,21)(17,22)(19,24)(20,23)(26,31)(28,35)(29,37)(34,39);(2,3)(6,7)(10,15)(12,17)(16,18)(19,23)(26,31)(27,33)(28,35)(36,40);(1,2,5,3)(4,6,8,7)(25,26)(27,29)(28,32)(30,31)(33,37)(34,36)(35,38)(39,40);(9,11,14,21)(10,16,15,18)(12,19,17,23)(13,20,22,24)(25,27)(26,29)(28,34)(30,33)(31,37)(32,36)(35,39)(38,40);(1,3,5,2)(4,7,8,6)(9,12,14,17)(10,13,15,22)(11,19,21,23)(16,20,18,24)(25,28,30,35)(26,32,31,38)(27,34,33,39)(29,36,37,40);(1,4)(2,6)(3,7)(5,8)(9,13)(10,17)(11,20)(12,15)(14,22)(16,23)(18,19)(21,24)(25,29)(26,27)(28,36)(30,37)(31,33)(32,34)(35,40)(38,39)Polycyclic presentation:
2^9|c^2=hi,d^2=g,e^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^b=egi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8017126);