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 | |
4419 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)(3,8,4,6)(9,10,14,15)(11,16,18,12)(13,21,23,17)(19,20,24,22)(25,26,30,31)(27,29,33,37)(28,32,35,38)(34,36,39,40);(3,4)(6,8)(10,15)(12,16)(17,21)(20,22)(28,35)(32,38)(34,39)(36,40);(1,3,5,4)(2,6,7,8)(10,15)(11,18)(17,21)(19,24)(26,31)(27,33)(32,38)(34,39);(1,2)(3,8)(4,6)(5,7)(9,11)(10,16)(12,15)(13,19)(14,18)(17,22)(20,21)(23,24)(25,27)(26,29)(28,34)(30,33)(31,37)(32,36)(35,39)(38,40);(1,4,5,3)(2,6,7,8)(9,12,14,16)(10,11,15,18)(13,22,23,20)(17,24,21,19)(25,28)(26,32)(27,34)(29,36)(30,35)(31,38)(33,39)(37,40);(1,2)(3,6)(4,8)(5,7)(9,13,14,23)(10,17,15,21)(11,19,18,24)(12,20,16,22)(25,29)(26,27)(28,36)(30,37)(31,33)(32,34)(35,40)(38,39)Polycyclic presentation:
2^9|a^2=h,c^2=i,e^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dh,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9627437);