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 | |
4426 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(3,7)(6,8)(9,10)(11,19)(12,15)(13,22)(14,20)(16,21)(17,23)(18,24)(26,31)(28,37)(29,36)(32,38);(2,5)(6,8)(10,12)(13,23)(16,19)(18,20)(26,31)(27,35)(28,37)(34,40);(1,2)(3,6)(4,5)(7,8)(9,11)(10,16)(12,19)(13,20)(14,17)(15,21)(18,23)(22,24)(25,26)(27,32)(28,33)(29,34)(30,31)(35,38)(36,40)(37,39);(9,12,15,10)(11,19,21,16)(13,22,23,17)(14,20,24,18)(25,27)(26,32)(28,36)(29,37)(30,35)(31,38)(33,40)(34,39);(1,2)(3,6)(4,5)(7,8)(9,13)(10,17)(11,20)(12,22)(14,16)(15,23)(18,21)(19,24)(25,28,30,37)(26,33,31,39)(27,36,35,29)(32,40,38,34);(1,3,4,7)(2,6,5,8)(9,14,15,24)(10,18,12,20)(11,17,21,22)(13,16,23,19)(25,29,30,36)(26,34,31,40)(27,37,35,28)(32,39,38,33)Polycyclic presentation:
2^9|d^2=g,e^2=gh,f^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, 8017765);