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 | |
4489 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,4,6)(3,7,8,5)(10,15)(11,17)(12,19)(20,24)(26,31)(27,34)(33,38)(35,40);(1,3,4,8)(2,5,6,7)(9,10,14,15)(11,13,17,21)(12,16,19,22)(18,20,23,24)(25,26,30,31)(27,29,34,32)(28,33,37,38)(35,36,40,39);(1,3)(2,5)(4,8)(6,7)(25,27)(26,29)(28,35)(30,34)(31,32)(33,36)(37,40)(38,39);(1,3)(2,5)(4,8)(6,7)(9,11)(10,13)(12,18)(14,17)(15,21)(16,20)(19,23)(22,24)(25,27)(26,32)(28,35)(29,31)(30,34)(33,39)(36,38)(37,40);(1,2)(3,5)(4,6)(7,8)(9,12)(10,16)(11,18)(13,20)(14,19)(15,22)(17,23)(21,24)(25,28,30,37)(26,33,31,38)(27,35,34,40)(29,36,32,39);(9,13)(10,17)(11,15)(12,20)(14,21)(16,23)(18,22)(19,24)(25,29,30,32)(26,34,31,27)(28,36,37,39)(33,40,38,35)Polycyclic presentation:
2^9|a^2=i,b^2=g,e^2=hi,f^2=hi,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, 8367894);