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 |
| prime radical | #43010 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4547 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)(3,8)(4,6)(5,7)(9,10,14,16)(11,15,18,12)(13,21,24,17)(19,20,22,23);(1,3,6,5)(2,7,4,8)(9,11)(10,15)(12,16)(13,22)(14,18)(17,20)(19,24)(21,23)(25,26,30,31)(27,33,36,32)(28,37,39,29)(34,40,38,35);(1,2,6,4)(3,7,5,8)(10,16)(11,18)(17,21)(19,22)(26,31)(29,37)(32,33)(35,40);(1,3,6,5)(2,7,4,8)(9,12,14,15)(10,11,16,18)(13,23,24,20)(17,22,21,19)(25,27,30,36)(26,32,31,33)(28,38,39,34)(29,35,37,40);(1,4,6,2)(3,7,5,8)(9,13,14,24)(10,17,16,21)(11,19,18,22)(12,20,15,23)(25,28)(26,29)(27,34)(30,39)(31,37)(32,40)(33,35)(36,38);(1,5,6,3)(2,7,4,8)(9,11)(10,15)(12,16)(13,19)(14,18)(17,23)(20,21)(22,24)(25,29)(26,28)(27,35)(30,37)(31,39)(32,38)(33,34)(36,40)Polycyclic presentation:
2^9|a^2=gh,b^2=hi,c^2=g,d^2=gi,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,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, 8300665);