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 | |
4698 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)(3,5)(4,8)(6,7)(9,10)(11,13)(12,16)(14,15)(17,18)(19,22)(20,24)(21,23);(2,7)(3,8)(9,11)(10,13)(12,19)(14,18)(15,17)(16,22)(20,23)(21,24)(25,26,30,31)(27,29,33,38)(28,32,37,39)(34,36,35,40);(1,3,6,8)(2,4,7,5)(10,15)(11,18)(16,23)(19,24)(26,31)(29,38)(32,39)(36,40);(1,4,6,5)(2,3,7,8)(9,12)(10,16)(11,19)(13,22)(14,21)(15,23)(17,20)(18,24)(25,27,30,33)(26,29,31,38)(28,35,37,34)(32,40,39,36);(1,2)(3,4)(5,8)(6,7)(9,13,14,17)(10,11,15,18)(12,20,21,22)(16,24,23,19)(25,28,30,37)(26,32,31,39)(27,34,33,35)(29,36,38,40);(1,5,6,4)(2,8,7,3)(9,11)(10,17)(12,19)(13,15)(14,18)(16,20)(21,24)(22,23)(25,29)(26,33)(27,31)(28,36)(30,38)(32,35)(34,39)(37,40)Polycyclic presentation:
2^9|b^2=hi,c^2=g,d^2=ghi,e^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8809707);