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 | |
4674 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,6,7)(3,5,4,8)(9,10,14,15)(11,13,17,23)(12,16,21,24)(18,22,19,20)(25,26,31,34)(27,30,33,39)(28,29,32,38)(35,36,37,40);(2,7)(3,4)(9,11)(10,13)(12,19)(14,17)(15,23)(16,20)(18,21)(22,24)(25,27)(26,30)(28,35)(29,36)(31,33)(32,37)(34,39)(38,40);(1,3,6,4)(2,8,7,5)(10,15)(11,17)(16,24)(18,19)(27,33)(30,39)(35,37)(36,40);(1,4,6,3)(2,5,7,8)(9,12)(10,16)(11,18)(13,22)(14,21)(15,24)(17,19)(20,23)(25,28)(26,29)(27,35)(30,36)(31,32)(33,37)(34,38)(39,40);(1,5)(2,3)(4,7)(6,8)(9,13)(10,11)(12,20)(14,23)(15,17)(16,19)(18,24)(21,22)(25,29)(26,32)(27,36)(28,34)(30,37)(31,38)(33,40)(35,39);(9,11)(10,13)(12,18)(14,17)(15,23)(16,22)(19,21)(20,24)(25,30,31,39)(26,33,34,27)(28,36,32,40)(29,37,38,35)Polycyclic presentation:
2^9|a^2=i,c^2=g,d^2=g,f^2=hi,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, 8231358);