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 | |
4605 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)(3,7)(4,6)(5,8)(10,12)(11,17)(16,20)(19,21)(25,26,30,31)(27,28,35,32)(29,33,38,39)(34,36,40,37);(1,2,4,6)(3,5,8,7)(9,10,14,12)(11,15,17,18)(13,16,22,20)(19,23,21,24)(26,31)(28,32)(33,39)(36,37);(1,2)(3,5)(4,6)(7,8)(25,27)(26,32)(28,31)(29,34)(30,35)(33,37)(36,39)(38,40);(1,3)(2,5)(4,8)(6,7)(9,11)(10,15)(12,18)(13,21)(14,17)(16,24)(19,22)(20,23)(25,28,30,32)(26,27,31,35)(29,37,38,36)(33,40,39,34);(9,12,14,10)(11,18,17,15)(13,20,22,16)(19,24,21,23);(9,13,14,22)(10,16,12,20)(11,19,17,21)(15,23,18,24)(25,29,30,38)(26,33,31,39)(27,34,35,40)(28,36,32,37)Polycyclic presentation:
2^9|a^2=h,b^2=gh,d^2=h,e^2=ghi,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,f^d=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8366311);