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 | |
4462 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)(9,10,14,16)(11,15,18,12)(13,17,22,23)(19,21,24,20);(1,3,4,8)(2,5,6,7)(9,11)(10,15)(12,16)(13,19)(14,18)(17,21)(20,23)(22,24)(25,26)(27,34)(28,31)(29,32)(30,33)(35,39)(36,37)(38,40);(1,3)(2,5)(4,8)(6,7)(10,16)(11,18)(17,23)(19,24)(25,27,30,32)(26,29,33,34)(28,36,38,35)(31,39,40,37);(1,3)(2,5)(4,8)(6,7)(9,12)(10,11)(13,21)(14,15)(16,18)(17,24)(19,23)(20,22)(25,26)(27,29)(28,31)(30,33)(32,34)(35,37)(36,39)(38,40);(1,2)(3,5)(4,6)(7,8)(9,11)(10,12)(13,19)(14,18)(15,16)(17,20)(21,23)(22,24)(25,28,30,38)(26,31,33,40)(27,35,32,36)(29,37,34,39);(9,13)(10,17)(11,19)(12,20)(14,22)(15,21)(16,23)(18,24)(25,29)(26,32)(27,33)(28,37)(30,34)(31,36)(35,40)(38,39)Polycyclic presentation:
2^9|a^2=h,b^2=g,c^2=hi,e^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^c=egi,f^d=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8442199);