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 | |
4550 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,6)(3,4,7,8)(9,10)(11,17)(12,21)(13,23)(14,16)(15,20)(18,19)(22,24)(25,26,30,32)(27,33,35,28)(29,31,38,37)(34,39,40,36);(1,3)(2,4)(5,7)(6,8)(9,11)(10,15)(12,22)(13,24)(14,20)(16,17)(18,23)(19,21)(25,27,30,35)(26,28,32,33)(29,34,38,40)(31,36,37,39);(2,6)(3,7)(10,16)(13,21)(15,17)(19,24)(25,28,30,33)(26,27,32,35)(29,36,38,39)(31,34,37,40);(3,7)(4,8)(9,12)(10,13)(11,18)(14,23)(15,19)(16,21)(17,24)(20,22)(25,26)(27,28)(29,37)(30,32)(31,38)(33,35)(34,39)(36,40);(1,2)(3,8)(4,7)(5,6)(9,13)(10,12)(11,19)(14,21)(15,18)(16,23)(17,22)(20,24)(25,28)(26,27)(29,36)(30,33)(31,34)(32,35)(37,40)(38,39);(1,4,5,8)(2,7,6,3)(9,11)(10,17)(12,22)(13,19)(14,20)(15,16)(18,23)(21,24)(25,29,30,38)(26,31,32,37)(27,34,35,40)(28,36,33,39)Polycyclic presentation:
2^9|a^2=gh,b^2=ghi,c^2=ghi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8514704);