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 | |
4487 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,4)(6,7)(9,10)(11,19)(12,21)(13,15)(14,18)(16,20)(17,23)(22,24);(1,2,5,4)(3,7,8,6)(9,11,15,20)(10,16,13,19)(12,22,23,14)(17,24,21,18)(25,26,30,31)(27,32,36,39)(28,33,37,29)(34,35,40,38);(1,3,5,8)(2,6,4,7)(10,13)(16,19)(17,21)(18,24)(26,31)(28,37)(32,39)(34,40);(1,4,5,2)(3,7,8,6)(9,12,15,23)(10,17,13,21)(11,14,20,22)(16,18,19,24)(25,27)(26,32)(28,34)(29,38)(30,36)(31,39)(33,35)(37,40);(9,13,15,10)(11,19,20,16)(12,21,23,17)(14,24,22,18)(25,28)(26,33)(27,34)(29,31)(30,37)(32,35)(36,40)(38,39);(9,14)(10,18)(11,12)(13,24)(15,22)(16,17)(19,21)(20,23)(25,29,30,33)(26,28,31,37)(27,35,36,38)(32,40,39,34)Polycyclic presentation:
2^9|b^2=hi,c^2=gh,d^2=g,e^2=h,f^2=ghi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^c=egi,f^d=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8456705);