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 | |
4338 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,6)(3,4,8,7)(10,11)(15,17)(16,18)(22,23);(2,6)(4,7)(9,10,14,11)(12,17,20,15)(13,16,21,18)(19,23,24,22)(25,26,30,31)(27,29,33,38)(28,35,37,32)(34,40,39,36);(1,3,5,8)(2,7,6,4)(13,21)(16,18)(19,24)(22,23)(26,31)(29,38)(32,35)(36,40);(1,4)(2,3)(5,7)(6,8)(9,11,14,10)(12,17,20,15)(13,18,21,16)(19,23,24,22)(25,27,30,33)(26,29,31,38)(28,34,37,39)(32,36,35,40);(1,4)(2,3)(5,7)(6,8)(9,12,14,20)(10,15,11,17)(13,19,21,24)(16,22,18,23)(25,28,30,37)(26,32,31,35)(27,34,33,39)(29,36,38,40);(1,3)(2,4)(5,8)(6,7)(9,13,14,21)(10,16,11,18)(12,19,20,24)(15,22,17,23)(25,29)(26,33)(27,31)(28,36)(30,38)(32,39)(34,35)(37,40)Polycyclic presentation:
2^9|a^2=h,b^2=ghi,c^2=h,d^2=ghi,e^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,f^c=fgi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8137899);