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 | |
4340 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,6)(3,8)(9,10)(11,15)(12,20)(13,17)(14,18)(16,21)(19,22)(23,24)(25,26)(27,33)(28,31)(29,32)(30,35)(34,39)(36,38)(37,40);(1,2)(3,7)(4,8)(5,6)(9,11)(10,15)(12,17)(13,20)(14,19)(16,24)(18,22)(21,23)(25,26,29,32)(27,30,35,33)(28,31,36,38)(34,37,40,39);(1,2)(3,7)(4,8)(5,6)(11,19)(12,21)(15,22)(16,20)(25,26)(27,30)(28,31)(29,32)(33,35)(34,37)(36,38)(39,40);(2,6)(4,7)(9,12,14,21)(10,16,18,20)(11,17,19,23)(13,15,24,22)(25,27)(26,30)(28,34)(29,35)(31,37)(32,33)(36,40)(38,39);(1,3)(2,7)(4,6)(5,8)(9,10)(11,15)(12,16)(13,23)(14,18)(17,24)(19,22)(20,21)(25,28,29,36)(26,31,32,38)(27,34,35,40)(30,37,33,39);(1,4,5,7)(2,8,6,3)(9,13,14,24)(10,17,18,23)(11,20,19,16)(12,22,21,15)Polycyclic presentation:
2^9|b^2=gh,d^2=ghi,e^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,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, 8657841);