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 | |
4542 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(3,7)(5,8)(9,10)(11,15)(12,17)(13,19)(14,16)(18,20)(21,23)(22,24)(25,26,31,35)(27,29,33,32)(28,30,34,39)(36,38,40,37);(1,2,4,6)(3,5,7,8)(9,11)(10,15)(12,19)(13,17)(14,20)(16,18)(21,22)(23,24)(25,27,31,33)(26,32,35,29)(28,36,34,40)(30,37,39,38);(1,3)(2,5)(4,7)(6,8)(10,16)(11,20)(17,24)(19,21)(25,28,31,34)(26,30,35,39)(27,36,33,40)(29,38,32,37);(1,2)(3,5)(4,6)(7,8)(9,12)(10,17)(11,21)(13,18)(14,22)(15,23)(16,24)(19,20)(25,26)(27,32)(28,30)(29,33)(31,35)(34,39)(36,37)(38,40);(1,3)(2,5)(4,7)(6,8)(9,11)(10,18)(12,21)(13,17)(14,20)(15,16)(19,22)(23,24)(25,29,31,32)(26,33,35,27)(28,37,34,38)(30,36,39,40);(9,13)(10,19)(11,17)(12,15)(14,23)(16,21)(18,22)(20,24)(25,30)(26,34)(27,37)(28,35)(29,36)(31,39)(32,40)(33,38)Polycyclic presentation:
2^9|a^2=g,b^2=ghi,c^2=g,e^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^c=egi,f^d=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8424556);