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 | |
4448 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)(3,7)(4,5)(6,8)(9,10,14,11)(12,17,20,15)(13,18,21,16)(19,22,24,23)(25,26,30,31)(27,29,33,39)(28,35,36,32)(34,40,37,38);(2,5)(6,7)(10,11)(15,17)(16,18)(22,23)(26,31)(27,33)(28,36)(38,40);(1,2)(3,6)(4,5)(7,8)(10,11)(13,21)(15,17)(19,24)(27,33)(29,39)(34,37)(38,40);(1,3,4,8)(2,6,5,7)(9,11,14,10)(12,15,20,17)(13,18,21,16)(19,22,24,23)(25,27,30,33)(26,29,31,39)(28,37,36,34)(32,40,35,38);(9,12,14,20)(10,15,11,17)(13,19,21,24)(16,22,18,23)(25,28,30,36)(26,32,31,35)(27,34,33,37)(29,38,39,40);(9,13,14,21)(10,16,11,18)(12,19,20,24)(15,22,17,23)(25,29)(26,33)(27,31)(28,38)(30,39)(32,37)(34,35)(36,40)Polycyclic presentation:
2^9|a^2=gi,d^2=g,e^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=di,e^a=egi,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,d^c=dgi,f^c=fgi,e^d=egi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8842013);