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:
(9,10,15,16)(11,13,12,17)(14,18,22,23)(19,21,20,24)(25,26,30,29)(27,31,35,38)(28,36,37,32)(33,40,34,39);(1,2)(3,8)(4,6)(5,7)(10,16)(11,12)(18,23)(19,20)(25,27)(26,31)(28,34)(29,38)(30,35)(32,40)(33,37)(36,39);(1,3,5,6)(2,4,7,8)(9,11,15,12)(10,17,16,13)(14,19,22,20)(18,24,23,21)(25,28)(26,32)(27,33)(29,36)(30,37)(31,39)(34,35)(38,40);(1,2)(3,4)(5,7)(6,8)(9,12,15,11)(10,13,16,17)(14,20,22,19)(18,21,23,24)(25,29,30,26)(27,31,35,38)(28,36,37,32)(33,39,34,40);(1,2)(3,4)(5,7)(6,8)(9,13)(10,11)(12,16)(14,21)(15,17)(18,19)(20,23)(22,24)(25,28)(26,32)(27,34)(29,36)(30,37)(31,40)(33,35)(38,39);(1,4)(2,6)(3,7)(5,8)(9,14,15,22)(10,18,16,23)(11,19,12,20)(13,21,17,24)(25,27)(26,31)(28,33)(29,38)(30,35)(32,39)(34,37)(36,40)Polycyclic presentation:
2^9|a^2=h,c^2=ghi,d^2=h,f^2=g,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, 8571981);