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 | |
4523 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)(11,16,12,21)(13,24,18,14)(19,23,22,20)(25,26,31,32)(27,34,37,28)(29,30,39,33)(35,38,40,36);(1,3,5,8)(2,6,7,4)(9,11)(10,16)(12,15)(13,19)(14,20)(17,21)(18,22)(23,24)(25,27,31,37)(26,28,32,34)(29,35,39,40)(30,36,33,38);(1,3)(2,6)(4,7)(5,8)(10,17)(11,12)(14,24)(19,22)(26,32)(28,34)(30,33)(36,38);(2,7)(3,8)(9,12,15,11)(10,16,17,21)(13,19,18,22)(14,23,24,20)(25,28)(26,27)(29,36)(30,35)(31,34)(32,37)(33,40)(38,39);(1,3)(2,4)(5,8)(6,7)(9,13,15,18)(10,14,17,24)(11,19,12,22)(16,20,21,23)(25,29)(26,33)(27,35)(28,38)(30,32)(31,39)(34,36)(37,40);(1,4)(2,3)(5,6)(7,8)(9,14)(10,18)(11,20)(12,23)(13,17)(15,24)(16,22)(19,21)(25,30,31,33)(26,29,32,39)(27,36,37,38)(28,35,34,40)Polycyclic presentation:
2^9|a^2=gi,b^2=hi,d^2=gh,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, 8841775);