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 | |
4602 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,4)(3,7,8,6)(9,10)(11,15)(12,21)(13,16)(14,18)(17,20)(19,24)(22,23)(26,31)(27,35)(28,36)(34,38);(1,3)(2,6)(4,7)(5,8)(9,11)(10,15)(12,22)(13,19)(14,20)(16,24)(17,18)(21,23)(25,26)(27,32)(28,37)(29,34)(30,31)(33,36)(35,40)(38,39);(9,12,14,13)(10,16,18,21)(11,19,20,22)(15,23,17,24)(25,27,30,35)(26,32,31,40)(28,29,36,39)(33,38,37,34);(2,4)(6,7)(9,10)(11,17)(12,16)(13,21)(14,18)(15,20)(19,24)(22,23)(25,28)(26,33)(27,29)(30,36)(31,37)(32,38)(34,40)(35,39);(1,3)(2,7)(4,6)(5,8)(9,11)(10,17)(12,19)(13,22)(14,20)(15,18)(16,24)(21,23)(25,26)(27,32)(28,33)(29,38)(30,31)(34,39)(35,40)(36,37);(1,4,5,2)(3,7,8,6)(9,13,14,12)(10,16,18,21)(11,19,20,22)(15,24,17,23)(25,29)(26,34)(27,36)(28,35)(30,39)(31,38)(32,33)(37,40)Polycyclic presentation:
2^9|a^2=ghi,c^2=h,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^d=fgi,f^e=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8570162);