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 | |
4585 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10)(11,18)(12,20)(13,15)(14,16)(17,22)(19,23)(21,24)(25,26)(27,34)(28,31)(29,37)(30,33)(32,38)(35,39)(36,40);(1,2)(3,6)(4,5)(7,8)(9,11,14,13)(10,15,16,18)(12,21,22,19)(17,24,20,23)(27,28)(31,34)(35,36)(39,40);(2,5)(6,8)(10,16)(15,18)(17,20)(23,24)(25,27,30,28)(26,31,33,34)(29,35,38,36)(32,39,37,40);(3,7)(6,8)(9,12,14,22)(10,17,16,20)(11,19,13,21)(15,23,18,24)(25,28,30,27)(26,31,33,34)(29,35,38,36)(32,40,37,39);(1,3)(2,6)(4,7)(5,8)(9,13,14,11)(10,18,16,15)(12,19,22,21)(17,23,20,24)(25,29)(26,32)(27,35)(28,36)(30,38)(31,39)(33,37)(34,40);(1,2)(3,6)(4,5)(7,8)(9,10)(11,15)(12,17)(13,18)(14,16)(19,23)(20,22)(21,24)(25,26)(27,34)(28,31)(29,32)(30,33)(35,40)(36,39)(37,38)Polycyclic presentation:
2^9|b^2=g,c^2=gh,d^2=h,e^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8180287);