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 | |
4501 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,5)(6,8)(9,10,14,16)(11,19,21,15)(12,17,22,20)(13,23,24,18)(25,26,30,28)(27,31,33,35)(29,32,38,36)(34,39,37,40);(9,11,14,21)(10,15,16,19)(12,18,22,23)(13,17,24,20)(25,27)(26,31)(28,35)(29,37)(30,33)(32,40)(34,38)(36,39);(1,2)(3,6)(4,5)(7,8)(10,16)(12,22)(15,19)(18,23)(27,33)(31,35)(34,37)(39,40);(1,3,4,7)(2,6,5,8)(9,11)(10,15)(12,18)(13,20)(14,21)(16,19)(17,24)(22,23)(25,28,30,26)(27,31,33,35)(29,36,38,32)(34,39,37,40);(1,3)(2,6)(4,7)(5,8)(9,12)(10,17)(11,18)(13,19)(14,22)(15,24)(16,20)(21,23)(25,27)(26,31)(28,35)(29,34)(30,33)(32,39)(36,40)(37,38);(1,2)(3,6)(4,5)(7,8)(9,13,14,24)(10,18,16,23)(11,20,21,17)(12,19,22,15)(25,29)(26,32)(27,34)(28,36)(30,38)(31,39)(33,37)(35,40)Polycyclic presentation:
2^9|a^2=gi,b^2=g,d^2=ghi,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,f^b=fgi,e^c=egi,e^d=eghi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10259762);