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 | |
4567 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,3)(4,7,8,6)(9,10)(11,17)(12,16)(13,22)(14,18)(15,21)(19,24)(20,23)(25,26,30,31)(27,28,35,32)(29,33,39,40)(34,36,38,37);(2,3)(6,7)(9,11)(10,15)(12,19)(13,20)(14,21)(16,23)(17,18)(22,24)(26,31)(28,32)(33,40)(36,37);(1,3,5,2)(4,7,8,6)(9,12,14,13)(10,16,18,22)(11,19,21,20)(15,23,17,24)(25,27)(26,32)(28,31)(29,34)(30,35)(33,37)(36,40)(38,39);(1,4,5,8)(2,6,3,7)(9,13,14,12)(10,16,18,22)(11,20,21,19)(15,23,17,24)(25,28,30,32)(26,27,31,35)(29,37,39,36)(33,38,40,34);(9,11)(10,17)(12,19)(13,20)(14,21)(15,18)(16,24)(22,23)(25,27)(26,32)(28,31)(29,38)(30,35)(33,36)(34,39)(37,40);(9,10)(11,15)(12,16)(13,22)(14,18)(17,21)(19,23)(20,24)(25,29,30,39)(26,33,31,40)(27,34,35,38)(28,36,32,37)Polycyclic presentation:
2^9|a^2=hi,c^2=gh,d^2=g,f^2=h,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, 8570145);