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 | |
4566 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,7)(3,6,8,4)(9,10,15,17)(11,19,22,16)(12,13,21,23)(14,24,20,18)(25,26,30,33)(27,34,35,31)(28,38,32,29)(36,40,39,37);(1,3,5,8)(2,6,7,4)(9,11,15,22)(10,16,17,19)(12,14,21,20)(13,18,23,24)(27,35)(31,34)(36,39)(37,40);(1,3)(2,6)(4,7)(5,8)(10,17)(12,21)(14,20)(16,19)(25,27,30,35)(26,31,33,34)(28,36,32,39)(29,40,38,37);(2,7)(3,8)(9,12,15,21)(10,13,17,23)(11,20,22,14)(16,24,19,18)(25,27)(26,31)(28,39)(29,37)(30,35)(32,36)(33,34)(38,40);(1,3)(2,4)(5,8)(6,7)(9,13)(10,12)(11,18)(14,16)(15,23)(17,21)(19,20)(22,24)(25,28,30,32)(26,29,33,38)(27,36,35,39)(31,40,34,37);(1,4)(2,3)(5,6)(7,8)(9,14,15,20)(10,18,17,24)(11,21,22,12)(13,16,23,19)(25,29)(26,32)(27,37)(28,33)(30,38)(31,36)(34,39)(35,40)Polycyclic presentation:
2^9|a^2=gi,b^2=hi,c^2=gh,d^2=g,e^2=gh,f^2=g,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, 8841734);