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 | |
4578 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,4)(6,7)(9,10)(11,13)(12,20)(14,16)(15,19)(17,22)(18,23)(21,24)(25,26,30,29)(27,33,35,31)(28,36,37,32)(34,38,40,39);(1,2,5,4)(3,7,8,6)(9,11)(10,15)(12,18)(13,16)(14,19)(17,23)(20,21)(22,24)(25,27,30,35)(26,31,29,33)(28,34,37,40)(32,38,36,39);(1,3,5,8)(2,6,4,7)(10,16)(13,15)(17,20)(21,23)(25,27)(26,31)(28,34)(29,33)(30,35)(32,38)(36,39)(37,40);(1,4,5,2)(3,7,8,6)(9,11)(10,15)(12,18)(13,16)(14,19)(17,23)(20,21)(22,24)(25,27)(26,31)(28,34)(29,33)(30,35)(32,38)(36,39)(37,40);(9,12,14,22)(10,17,16,20)(11,18,19,24)(13,21,15,23)(25,28,30,37)(26,32,29,36)(27,34,35,40)(31,38,33,39);(9,13,14,15)(10,11,16,19)(12,21,22,23)(17,18,20,24)(25,29,30,26)(27,33,35,31)(28,36,37,32)(34,39,40,38)Polycyclic presentation:
2^9|a^2=ghi,b^2=i,c^2=gh,d^2=gh,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^c=fgi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8092031);